unit 3 – quadratic & polynomial modeling - ligon nc math...
TRANSCRIPT
!
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
"#$%&!!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""!
!
!
!
!
!
Common Core Math 3 Unit 3 – Quadratic & Polynomial Modeling
A P E X H I G H S C H O O L
1501 L A U R A D U N C A N R O A D
A P E X , N C 27502
!"##"$%!"&'%()*+%,%%
-$.*%,%/0)1&)*.23%4%5"67$"#.)63%
8)7% 8)*'% 9"#':"&;%
<%% %
=%% %
,%% %
>%% %
?%% %
@%% %
A%% %
B%% %
C%% %
<D%% %
!!"
" "
!#"
" "
!$"
" "
!%"
" "
!&"
" "
"
!"##"$%!"&'%()*+%,%-%.$/*%0,%12)3&)*/45%)$3%6"78$"#/)75%
9":/45%/$%*+/5%2$/*;%
! 12)3&)*/4%<2$4*/"$5%
o !"#$%#&%'#$%'()&")*'+,&-.'
o /$3''
o !,45)'67'+#8",&2$39'"#:2$3'"1)'.;<#&)'&,,"9';<#%&#"28'=,&-<4#9'3$39'#$%'
8,-04)"2$3'"1)'.;<#&)'
o >)&25)';<#%&#"28'+,&-<4#'67'8,-04)"2$3'"1)'.;<#&)'
o ?,-04)*'$<-6)&.'#$%',0)&#"2,$.'@2"1'8,-04)*'$<-6)&.'
o A<#%&#"28'-,%)42$3'
o >)+2$2"2,$',+';<#%&#"28'@2"1'+,8<.'#$%'%2&)8"&2*'
o >)&25)';<#%&#"28');<#"2,$'+&,-'+,8<.'#$%'%2&)8"&2*'
o !<-'#$%'0&,%<8"',+'&,,".'BC,$,&.D'
! 6"78$"#/)7%<2$4*/"$5%
o >)+2$2"2,$',+'#'0,47$,-2#4'+<$8"2,$%
o !"#$%#&%'+,&-9'#$%'84#..2+7'67'%)3&))'#$%'E',+'")&-.%
o >252%2$3'@2"1'4,$3'%252.2,$'#$%'.7$"1)"28'%252.2,$%
o F$%'6)1#52,&'#$%'.:)"812$3'3.%
o G)&,.9'&,,".9'.,4<"2,$.9'*H2$")&8)0".%
o I)-#2$%)&'#$%'=#8",&'J1),&)-.%
o I#"2,$#4'I,,"'J1),&)-'BC,$,&.D%
Students will be able to . . .
" !,45)';<#%&#"28');<#"2,$.'@2"1'&)#4'8,)++282)$".'"1#"'1#5)'8,-04)*'.,4<"2,$.K!
" ?,-04)")'"1)'.;<#&)'2$'#';<#%&#"28')*0&)..2,$'",'&)5)#4'"1)'-#*2-<-',&'-2$2-<-'5#4<)',+'
"1)'+<$8"2,$'2"'%)+2$).K'!
" !,45)'A<#%&#"28.'67'+#8",&2$39'3$39'2$.0)8"2,$9'.;<#&)'&,,".9'8,-04)"2$3'"1)'.;<#&)9'
#$%';<#%&#"28'+,&-<4#K'!
" >)&25)'"1)';<#%&#"28'+,&-<4#'<.2$3'8,-04)"2$3'"1)'.;<#&)K'!
" L%%9'.<6""9'#$%'-<4"2047'8,-04)*'$<-6)&.'!
" >)&25)'"1)');<#"2,$',+'#'0#,4#'325)$'#'+,8<.'#$%'%2&)8"&2*K'!
" M&2")'#';<#%&#"28');<#"2,$'+&,-'2".'&,,".K'BC,$,&.D!
" ?&)#")');<#"2,$.'2$'"@,'5#&2#64).'",'&)0&).)$"'&)4#"2,$.120.'6)"@))$';<#$"2"2).K!
" N$%)&."#$%'"1)'&)4#"2,$.120'6)"@))$'O)&,.'#$%'+#8",&.',+'0,47$,-2#4.K'!
" >252%)'0,47$,-2#4.'<.2$3'4,$3'#$%'.7$"1)"28'%252.2,$K!
" /'0,47$,-2#4'+<$8"2,$.9'2%)$"2+72$3'O)&,.'@1)$'.<2"#64)'+#8",&2O#"2,$.'#&)'#5#24#64)9'#$%'
.1,@2$3')$%'6)1#52,&K''
" N.)'"1)'&)-#2$%)&'"1),&)-'",'+#8",&'0,47$,-2#4.',+'"12&%'%)3&))'B,&'1231)&'P'C,$,&.DK'
" N.)'"1)'&#"2,$#4'&,,"'"1),&)-'",'+#8",&'0,47$,-2#4.'BC,$,&.D'
VOCABULARY
! !"!"#$%#&'()*"+(&',+)#$"%"&'()*#+("+&"*,-"&+./
f (x) = ax2
+ bx + c )0,-.-"
a,b,&c "
%.-".-%1"('/2-.$"%(3)
a ! 0 -"
! .&#+$#%$)*,%/"+&"%"4'%3.%*#)"-4'%*#+(
y = ax2
+ bx + c "! 01%&12)*,%/)+&"%"4'%3.%*#)"-4'%*#+()
! 5,-"31%&12)+&"%"4'%3.%*#)"-4'%*#+("#$"*,-"/%6#/'/"+."/#(#/'/"7+#(*8)
! 41%,56)5,7"&',+56)%,,&5)%.-"*,-"9%1'-$"+&"6"*,%*"/%:-";<=8"
! 28'+&1%(19&5)%.-"*,-"7+#(*$"0,-.-"%">.%7,").+$$-$"*,-"6?%6#$@"*,-".-%1"9%1'-$"+&"6"*,%*"
/%:-";<=8"
! 5,-"!"#$%#&'()*,%/"7#"#$""
x =-b ± b
2- 4ac
2a !!A*")%("2-"'$-3"*+"$+19-"%"4'%3.%*#)"#("
$*%(3%.3"&+./8"
! !"(,/9712)+"/:1%"#$"%(;"('/2-."*,%*")%("2-"0.#**-("#("*,-"&+./""#));)):!6""0,-.-"%""%(3""
2"%.-".-%1"('/2-.$@"%(3"#<
-1 "8"
! 5,-"(,/9712)(,+<"=#&1)+&"!"B""#"#$"!"?""#"8)
! !""#$#"!%&)#$"%("-67.-$$#+("*,%*"#$"%".-%1"('/2-.@"%"9%.#%21-@"+."*,-"7.+3')*"+&".-%1"
('/2-.$"%(3"9%.#%21-$8""""
! !"'#&($#"!%&"#$"%"/+(+/#%1"+."*,-"$'/"+&"/+(+/#%1$8"
! !"'#&($#"!%&)*+$,-!#$"#$"%(;"&'()*#+("0#*,"%".'1-"*,%*")%("2-"0.#**-("#("*,-"&+./C"
$%&'()((!*&*(+(!*,-&
*,-(+(.((+(!-&(+(!/((((((0,-.-""%("D"=@"")+-&&#)#-(*$"E%(@"F@"%=G"%.-".-%1"
('/2-.$@"%(3"-67+(-(*$"%.-"(+((->%*#9-"#(*->-.$8"
! !(;"%1>-2.%#)"-67.-$$#+("#("*,-"&+./"!*&*(+(!*,-&
*,-(+(.((+(!-&(+(!/((#$")%11-3"%"'#&($#"!%&)
./'0.11!#$8"""
! 5,-"$1=%11"+&"%"7+1;(+/#%1"&E6G"#$"*,-",#>,-$*"-67+(-(*"+("*,-"9%.#%21-@"68""
! 5,-)&.%2!$3),#.**!,!.$-"+&"%"7+1;(+/#%1"#$"*,-")+-&&#)#-(*"+&"*,-"*-./"0#*,"*,-",#>,-$*"
3->.--8"
! A()1-%$2%02)*#0")*,-"*-./$"+&"%"7+1;(+/#%1"%.-"0.#**-("#("3-$)-(3#(>"+.3-."2;"3->.--8"
! >+$):1?#3',%"A$"*,-"2-,%9#+."+&"%">.%7,"%$"6"%77.+%),-$ +! or -! .
! @"7&'97'('&A"#$"*,-"('/2-."+&"*#/-$"%"H-.+"+))'.$8"
! .A+&?1&'()$'3'5',+"#$"%"$,+.*)'*"/-*,+3"&+."3#9#3#(>"%"7+1;(+/#%1"2;"%"1#(-%."&%)*+."+&"*,-"
&+./"%&(0(!'8"""
"
"
y = a(x - h)2
+ k
QUADRATIC REVIEW
Quadratic Function:
f (x) = ax2
+ bx + c where
a,b,&c are real
numbers and
a ! 0
! standard form
y = ax2
+ bx + c
" quadratic term: ax2, linear term: bx , constant term: c
! vertex form
y = a(x - h)2
+ k where
a,h,&k are real numbers and
a ! 0
! the graph is called a parabola
" If the “a” is positive (+), the parabola opens up.
" If the “a” is negative (–), the parabola opens down
! to graph a quadratic
" start with the parent graph (
y = x2
) and apply transformations
" make a T-chart
" use your calculator
! axis of symmetry
" in standard form
x = -b
2a
" in vertex form x=h
! vertex: the high point or low point of the graph
" in standard form
x = -b
2a is the x coord. of the vertex
" in vertex form the vertex is (h,k)
" the x coord. of the vertex is always the x value halfway between the x-
intercepts
" to find the y coord. plug in the value of the x coord. and solve for y
" use the calculator (2nd
Trace > minimum or maximum)
! x-intercepts: where the graph crosses the x-axis
" the real values of x that make y=0
" possible number of x-intercepts: 0, 1, 2
! zeros, solutions, roots: the values of x that make y=0
" may or may not be real
" real zeros, solutions, roots are x-intercepts
! quadratic equation –
ax2
+ bx + c = 0 , to solve:
" use the calculator (2nd
Trace > zero)
" when there is no linear term, set y = 0 and solve for x (take the ± square
root)
" by factoring, set each factor = 0 and solve for x
" standard form: use the quadratic formula
x =-b ± b
2- 4ac
2a
! discriminant is the value of
b2
- 4ac
"
b2
- 4ac = 0 - one real rational double root; vertex of parabola lies
on the x-axis
"
b2
- 4ac > 0 and a perfect square - two real rational roots;
parabola intersects x-axis twice
" - two real irrational
roots; parabola intersects x-axis twice
"
b2
- 4ac < 0 - no real roots, two complex conjugate/imaginary roots;
parabola does not intersect the x-axis
! Using the calculator:
Enter your quadratic into Y=
(Be sure to use X as your independent variable)
To Find a Vertex (Maximum/Minimum):
1. Enter equation in Y =
2. Use CALC menu (2nd
TRACE)
Choose #3: minimum or #4: maximum
3. Move curser left/right until it is to the left of the vertex (close to point). Press ENTER
4. Move curser left/right until it is to the right of the vertex (close to point). Press ENTER
5. Press ENTER to reveal vertex (max/min)
To Find Zeros/Roots/X-Intercepts:
1. Enter equation in Y =
2. Use CALC menu (2nd
TRACE) Choose #2: zero
3. Move curser left/right until it is to the left of the zero (close to point). Press ENTER
4. Move curser left/right until it is to the right of the zero (close to point). Press ENTER
5. Press ENTER to reveal zero
You will need to repeat for each zero.
b2
- 4ac > 0 and not a perfect square
Academic
FACTORING FLOW CHART
Check for a GCF.
Polynomial with 4 or more terms
Factor by Grouping
Addition or Subtraction?
Binomial Factors
Binomial (2 terms)
Trinomial (3 terms)
Factor into Binomial Factors
(Use any method or shortcut you’ve learned)
Binomial Factors
Addition Subtraction
Done
Check for Difference of Squares
a2-b2=(a+b)(a-b)
Done
Sum of Squares
PRIME
Honors
FACTORING FLOW CHART
Check for a GCF.
Polynomial with 4 or more terms
Factor by Grouping
Addition or Subtraction?
Binomial Factors
Binomial (2 terms)
Trinomial (3 terms)
Factor into Binomial Factors
(Use any method or shortcut you’ve learned)
Binomial Factors
Addition Subtraction
Done
Check for Difference of Squares or Cubes
a2-b2=(a+b)(a-b) a3-b3=(a-b)(a2+ab+b2
)
Done
Check for Sum of Cubes
a3+b3=(a+b)(a2-ab+b2)
Quadratics Worksheet 1
Graph: plot the vertex and 4 more points (2 on each side of vertex)
Parent Function y = x2 1. y = (x – 2)
2 + 1 2. y = x
2 +6x+5
3. y = –(x + 1)2 + 3 4. y = 2x
2 –12x +13 5. y = (x)
2 – 2
Put the following in standard form f(x) = ax2 + bx + c. Name the vertex and axis of symmetry!
6. f(x) = (x – 3)2 + 4 7. f(x) = (x + 1)
2 – 3 8. f(x) = 2(x – 4)
2 – 3
Solve by Graphing:
9.
Solve by Graphing:
Parent Function y = x2 10. y = (x – 3)
2 – 1 11. y = -x
2 – 4x
x- intercepts: __________ _____________ _____________
12. y = –3(x + 4)2 + 3 13. y = 2x
2 – 4x 14. y = (x – 5)
2 – 2
x- intercepts: __________ _____________ _____________
Name the vertex of the graph _______________
Name the axis of symmetry ________________
What are the x-intercepts? _________________
Write the equation ________________________
Solve by factoring:
15. x – 2x – 15 = 0 16. z – 5z = 0
17. x + 6x = -9 18. 3q – 7q = 20
18. 9y = 49 19.
2c2
- 24c + 54 = 0
20.
25x2
- 4 = 0 21.
25x2
- 30x + 9 = 0
Solve by taking the square root:
22.
5a2
-15 = 0 23.
3 x - 2( )2
= 24
24.
1
5x - 4( )
2
= 6 24.
3x2
+ 42 = 0
Quadratic Worksheet 2
Identify the quadratic term, the linear term, and the constant term for each function.
1. f(x) = x2 + 14x + 49 2. f(x) = -3(2x + 1)
2
Graph each function. Name the vertex and the axis of symmetry.
3. f(x) = x2 – 10x + 25 4. f(x) = (x + 4)
2 – 6 5. f(x) = -(x – 1)
2 + 4
Vertex: __________ _____________ _____________
axis of sym: __________ _____________ _____________
Solve (i.e. find the x-intercepts) by graphing.
6. f(x) = -(x + 5)2 + 1 7. f(x) = x
2 + 2x 8. f(x) = 2(x + 3)
2 – 8
Vertex: __________ _____________ _____________
x- intercepts: __________ _____________ _____________
Solve each equation. Remember to set equal to zero. If there is a linear term you can solve
some by factoring. If there is no linear term solve by taking the square root.
9. x2 – 4x – 12 = 0 10. x
2 – 16x + 64 = 0
11. x2 + 25 = 10x 12. 9z = 10z
2
13. 7x2 – 4x = 0 14. x
2 = 2x + 99
15. 5w2 – 35w + 60 = 0 16. 3x
2 + 24x + 45 = 0
17. 15m2 + 19m + 6 = 0 18. 4x
2 + 6 = 11x
19. 36x2 = 25 20. 12x
3 – 8x
2 = 15x
21. 6x3 = 5x
2 + 6 x 22. 9 = 64x
2
!"#$%&'(%&)*+,,$'-".,'/'0'!"#$%&'#%()*+$+*,-' ' ' ''
.-''"/'0'",'0'1/'0'1,'' ' ' ' 2-''3/2'0'2/,'0'.2/'0'4,'
'
'
'
4-''5(6'!'7('!'86'!'5' ' ' ' 8-''2/2,'0'5/,'!'/'!'4''
'
'
'
9-''8:'0'.2'0':2'0'4:' ' ' ' 5-'')2;'0');'!'.'!')''
'
'
'
<-''2"#'0'"='0'51#'0'41=' ' ' ' 3-''8&2>'!'3&>'!'4&'0'5'
'
'
'
7-''?4'!'5'0'2?'!'4?2'' ' ' ' .@-''4"'!'9"2'!'51'0'.@"1'
'
'
'
..-''2AB'!'A2B'!'5'0'4A' ' ' ' .2-''5#=2'!'3#='!'7='0'.2'
'
'
'
.4-''2+2C'!'.2+C'0'4+'!'.3''' ' ' .8-''4"#'0'41#'0'"='0'1='
'
'
'
.9-''2#/'0'#,'!'2=/'!'=,'' ' ' ' .5-''1/8'!'1,8'0'#/8'!'#,8'
'
'
'
.<-''&2'0'5&$'0'7$2'!'"2'!'2"1'!'12'' ' .3-''8/2'0'8/,'0',2'!'7"2'!'.2"$'!'8$2'
'
'
'
.7-''5/4'0'7/'!'8/2'!'5' ' ' ' 2@-''2/?'!'5/,'0'2,?'!'5,2'
'
!"#$%&'(%&)*+,,$'-".,'/'0'!"#$%&'#%()*+$+*,-'
' '
./-''01.'!'231'4'.5'' ' ' ' ..-''2.1.'4'/61'4'.'
'
'
'
.2-''51.'!'231'4'75'' ' ' ' .7-''/61.'4'731,'4'.5,.'
'
'
'
.5-''801.'4'/.1'!'7''' ' ' ' .6-''5#.'4'.3#'4'.3'
'
'
'
.9-''01.'!'7.1,'4'70,.' ' ' ' .:-''91.'!'9,.'
'
'
'
.0-''"7'!';7'' ' ' ' ' ' 23-''71.'4'67'
'
'
'
2/-''.1.'!'53'' ' ' ' ' 2.-''7:'!'2<.'
'
'
'
22-''".;'!'/6;5' ' ' ' ' 27-''7,.'!'/61.'
'
'
'
25-'':/=7'!'#7' ' ' ' ' 26-''".;2'!'.5;2#.'
'
'
'
29-''/61.'4'.71'4'0'' ' ' ' 2:-''1.'!'61'4'0'
'
'
'
20-''.51.'!'.31'4'7'' ' ' ' 73-'',7'!'/6'
'
!"#$%&'(%&)*+,,$'-".,'/'0'!"#$%&'#%()*+$+*,-' './%0%&12'
'
34-''#5'6'7'
'
'
'
38-''95'6':3'
'
'
'
35-''7;5'6'535,5'
'
'
'
33-''(5'!'48<''
'
'
'
3<-''=5'!'84:''
'
'
'
3:-''8>;5'!',5'
'
'
'
3>-''7;5'!'8>,5'
'
'
'
37-''7;:'6'8>,:'
'
'
'
3?-''8;5'!'<3''
'
'
'
<@-''4:;3'!'74'' ' ' '
' '
<4-'';3'!'4@;8'6'?'
'
'
'
<8-'';:'!'?;5'6'7'
'
'
'
<5-'';3'A'45;8'6'5:'
'
'
'
<3-'';3'!'5;8'!'3''
'
'
'
<<-'';3'6'<;8'!'5:''
'
'
'
<:-''8;3'6'>;8'!'3''
'
'
'
<>-''5;3'!'7;8'6'3'
'
'
'
<7-'':,3'!'8,8'!'3''
'
'
'
<?-''":'!'8"5'A'4<''
'
'
'
:@-''4<;8'!'43;,'!'7,8'
Quadratic Formula:
Solve using the Quadratic Formula
1.
3x2
+ 8x = 35 2.
3. 4.
5. 6.
Solve by taking the square root:
7. 3x2 = -81 7. 5x
2 + 18 = 3
8. (m – 2)2 = -16 10.
x =-b ± b
2- 4ac
2a
Complex Numbers
We do NOT get a real number when we take the square root of a
negative number. For example
-9 is not a real number because there is
no real number that can be squared to a get -9.
Imaginary numbers are used when there is a negative number under a
square root. “i” is used to signify an imaginary number. The reason for
the name "imaginary" numbers is that when these numbers were first
proposed several hundred years ago, people could not "imagine" such a
number.
i=
-1 so …
-4 =
-1! 4 =
i 4= 2i
i =
-1 i5 = i
9 = i
13 =
i2 = i
6 = i
10 = i
14 =
i3 = i
7 = i
11 = etc….
i4 = i
8 = i
12 =
To simplify imaginary numbers with an exponent greater than 3:
1) Divide the exponent by 4
2) The remainder becomes the new exponent
3) Simplify
Examples: i13
i12
i94
i27
To simplify the square root of a negative number:
1) pull out the i
2) simplify the radical
Examples:
If two square roots with negative numbers are being multiplied: pull out
the i BEFORE you multiply!
Examples:
-6 ! -10
-8 ! 2
Adding/Subtracting: combine like terms
Examples: (8 – 5i) + (2 + i) (4 + 7i) – (2 – 3i)
Multiplying with imaginary numbers: NEVER leave i2 in your answer!
Examples: (4 + 2i)(3 – 5i) (4 – i)(3 + 2i)
A complex number is any number that can be written in the
standard form a + bi, where a and b are real numbers, and i=
-1 .
! real numbers are complex numbers with b=0 ! pure imaginary numbers are complex numbers with a=0
Every complex number has a complex conjugate. The complex
conjugate of a + bi is a - bi . For example the conjugate of
3 + 5i is 3 – 5i.
What happens when you multiply conjugates?
Examples: (2 + i)(2 – i) (3 + 5i)(3 – 5i)
!"#$%&'()*#+&,-(.",/-0&&1((23#$%3456(
76(( 49! ( ( ( ( ( ( ( 86((
"
!!" ((
96(( 126 ! ( ( ( ( ( ( ( :6(( 123 !#! ((
;6(( 105 #! ( ( ( ( ( ( <6(( 72! ((
=6((42i ( ( ( ( ( ( ( >6((
91i ((
?6((@=(!(<3A(B(@?(B(773A( ( ( ( ( 7C6(( )413()85( ii $!$$ (
(776((D<@8(!(>3A(B(9@;(B(=3A( ( ( ( ( 786((:@=(!(3A(D(;@8(!(<3A(
(
796((@9(!(:3A8( ( ( ( ( ( ( 7:6((2)25( i$ (
(7;6((@<(!(:3A@<(B(:3A( ( ( ( ( ( 7<6(( )28)(28( ii $! (
(7=6((;@8(B(93A(B(<@>(!(;3A( ( ( ( ( 7>6((@:(B(93A@8(!(;3A@:(!(93A(
(E3FG(10&(HI%*&-("4('(IFG(5(4",(J03K0(&IK0(&L*I13"F(3-(1,*&6((
7?6((9'(!(;53(M(7;(!(8C3( ( ( ( ( 8C6((;'(B(=53(M(<(!(83(
(
2"%H&(&IK0(&L*I13"F(+5(1I/3FN(10&(-L*I,&(,""16((
876((F8(B(8;(M(C( ( ( ( ( ( 886((@#D8A8(B(7C(M(C(
(
(
896((<@5D9A8(B(:8(M(C( ( ( ( ( 8:6((:,8(B(<:(M(C(
(
2"%H&(&IK0(&L*I13"F(+5(*-3FN(10&(L*IG,I13K(4",#*%I6((
8;6((:F8(B(7C(M(78F( ( ( ( ( ( 8<6((/8(B(>/(M(8C(
(
(
(
8=6((:'8(B(>'(M(D7C( ( ( ( ( ( 8>6((:,8(B(78,(M(D>(
!"#$%&'()*+',&+-./01&++
!"#$%&'()*"#)"#+,)-,#*"$#.)/$#).#'#0$12$-(#.%&'1$#(1)"*3)'4#5#################################################5#
-'"#6$#$'.)47#.*48$/#67#('9)":#(,$#.%&'1$#1**(#*2#$'-,#.)/$;#
x2
- 6x + 9 = 25
(x - 3)2
= 25
(x - 3)2
= ± 25
(x - 3) = ±5
x = 3 ± 5
x = 8,-2
+
<,'(#.,*&4/#6$#'//$/#(*#2*13#'#0$12$-(#.%&'1$#(1)"*3)'4=#
23++
x2
+ 2x _____ = ( )2
+ + + + 43++
x2
+16x _____ = ( )2
+
+
53++
x2
+ 6x _____ = ( )2
+ + + + 63++
x2
- 8x _____ = ( )2
+
+
73++
x2
+10x _____ = ( )2
+ + + + 83++
x2
-12x _____ = ( )2
+
+
93++
x2
- 4x _____ = ( )2
+ + + + :3++
x2
+ 8x _____ = ( )2
+
+
+
>,$#3$(,*/#*2#.*48)":#67#?2*1-)":?#'#0$12$-(#.%&'1$#(1)"*3)'4#).#-'44$/#5;<=>7?@A9+?:7+
BCD2E7;#
&F2<=>7+GH##@*48$#AB#C#DA#C#E#F#G#
#
#
#
#
#
#
#
#
#
#
#
#
+
#
#
!
"#$%&'()*+!(,'!-./01'!0&-#!(02'-!0!./0310()4!)*!-(0*3013!5#1$!0*3!4#*6'1(-!)(!(#!6'1('7!
5#1$8!!!9*!(,'!'70$%&'!0:#6';!!!7<!=!>7!=!?!@!A!!!)-!4#*6'1('3!(#!B7!=!CD
<!!E!F!@!A8!
!
95!(,'!4#'55)4)'*(!#5!(,'!7<!('1$!)-!*#(!?G!3)6)3'!'04,!('1$!:A!(,0(!60&/'!(#!41'0('!0!&'03)*+!
4#'55)4)'*(!#5!#*'8!!!
H70$%&'!<;!!<%<!=!<I!@!>%!
!
!
!
!"#$%&$'())*+,-')./)$+01,'%&23)%4')567#"'8))
?8!7<!J!F7!J!C!@!I! ! ! ! ! <8!7
<!=!K7!J!L!@!I!
!
!
!
!
!
!
!
!
!
!
"#!!$%!&!"$!&!'!(!)! ! ! ! ! *#!$
%!+!'$!&!%!(!)!
!
!
!
!
!
!
!
!
!
!
!
!
'#!!%$%!&!,%$!+!-!(!)! ! ! ! ! .#!"$
%!+!,'$!&!,%!(!)!
!
!
!
!
!
!
!
!
!
!
!
!
!
/#!!$%!&!$!&!-!(!)! ! ! ! ! -#!$
%!+!%)$!+!%'!(!)!
!
!
!
!
!
!"#$%&'(&)"*+#%,-./&,0%&!1234%&5&6"4780%%,&
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17.
!"#$%"&'("&)*+,#+'$-&./#0*1+&23&4/051"'$67&'("&89*+#"&
!"#$"%&'"(%"()%)*+#"',-&&&&&&!"# $ %" $ & ' (&
.,&%/,01)%23%4,560)"'-7%"()%/*+#$)8%
Solving Quadratic Equations
Solve using any method.
1. 2x2 = 16
2. 4x2 + 8 = 0
3. 3x2 + 8x + 4 = 0
4. 9x2 + 15 = 0
5. 3x2 + 8 = 10
6. 2y2 + 2y – 24 = 0
7. b2 - 12b = 2b – 45
8. x2 = 8x + 20
9. 25x2 = -4
10. 5x2 + 6x – 12 = -4
11. 4x2 = 9
12. 2x2 + 12 = 0
13. 3x2 - 7x = 6
14. 2x2 = 12x – 16
15. x2 + 6x = 40
16. 15x2 = -10
17. x2 – 3x + 20 = 38
18. 15x2 + 8 = 5
19. 3n2 – 6n – 45 = 0
20. 5x2 – 12 = 18
21. 9x2 – 3x = 0
22. 3x2 – 8x = 0
23. 8x2 – 12 = -15
24. y2 - 7y = 30
25. x2 -7x + 10 = 0
26. 4x2 = 6x
27. 3x2 +4x – 12 = 3
28. 6x2 +17x + 5 = 0
29. 4y2 = -11y – 6
30. 6x2 = 3 - 7x
!"#$%&'$()*'"+,$*-$.**,/$$$ $ $ $ $ $ 01*&*)/2$!
"#$%&!'!()'*+',#-!%()',#.&/!!!!!!!'01!!2!30!!2!!-!!4!!5!!
!
!a
acbba
acbb2
42
,2
42
!!!!"! !
Sum of the roots of a quadratic equation = ab!
#
!! " !!! ""#
!""!!! !
!! ""#
!"$!! " !
!! ""# ! !! !
!! ""#
!"$ !!!
!"$ !!
"
Product of the roots of a quadratic equation = ac
#
!! " !!! ""#
!"%!!! !
!! ""#
!"$!!" ! !
!! ""# ! ! !! ! ""# ! #!! ! ""#$
""!$#
"!
!
!"#$*-$)**,/$$3$ ab!
$ ()*'"+,$*-$)**,/$$3$$ ac
$
6+#,%!'!4"%')%,5+$64"%,5*&!,7',!7'8!5&,67)%8$+*6--5+56&,/!'&*!,7%!9#$%&!+..,8:!
;:!!!<!!=!'!*.)3>%!+..,?! ! ! ! ! 1:!!@A!!B<!!!
!
!
<:!!4
3,4 ! ! ! ! ! ! C:!! 52 & !
!
!
@:!! i3& ! ! ! ! ! ! D:!! i25& !!!
!
!
E:!! 0,5
6! ! ! ! ! ! ! F:!! 32 i& !!!
!
!
G:!!3
1,2 ! ! ! ! ! ! ! ;5:!! 51,51 ii !!"! ! ! ! !
H7%!8.>),#.&8!.I!'!
()'*+',#-!%()',#.&!
I+.J!,7%!()'*+',#-!
I.+J)>'/!
Sum & Product of Roots Worksheet (Honors)
Solve each equation, then find the sum and product of the roots to check your solutions.
1. x2 – 7x + 4 = 0 2. x2 + 3x + 6 = 0
3. 2n2 + 5n + 6 = 0 4. 7x2 – 5x = 0
5. 4r2 – 9 = 0 6. –5x2 – x + 4 = 0
7. 3x2 + 8x = 3 8.
Write the quadratic equation, in standard form, that has the given roots
9. 7, -3 10.
11. 12.
13. 14. 7 – 2i , 7 + 2i
15. 8i, -8i 16.
17. 18.
Find k such that the number given is a root of the equation.
19. 7; 2x2 + kx – 21 = 0 20. –2; x2 – 13x + k = 0
!"#"$%&"'('!"#$%&$'(($)%*+#!$*+$'$)('+"$#,'#$'-"$#,"$!'."$/*!#'+0"$
&-%.$'$1*2"+$)%*+#$3)%*+,4$#%$'$1*2"+$(*+"$3-.#/*0#.14$$
! 5,"$2"-#"6$*!$'(7'8!$,'(&7'8$9"#7""+$#,"$&%0:!$'+/$#,"$/*-"0#-*6;$
o )$*!$#,"$/*!#'+0"$&-%.$#,"$&%0:!$#%$#,"$2"-#"6$'+/$#,"$/*!#'+0"$
&-%.$#,"$2"-#"6$#%$#,"$/*-"0#-*6$$
! 5,"$)'-'9%('$'(7'8!$0:-2"!$'7'8$&-%.$#,"$/*-"0#-*6;$
! 5,"$/*-"0#-*6$*!$)"-)"+/*0:('-$#%$#,"$'6*!$%&$!8.."#-8;$!
!!!<%-$)'-'9%('!$#,'#$%)"+$:)=$ $ $ $ <%-$)'-'9%('!$#,'#$%)"+$/%7+=$!
!!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
2%%3'"0'04/'5')%#6,'%)'04/'/7+"0.%89'8%0.*/'04"0:'
! 3,>?4$*!$#,"$2"-#"6$3!'."$'!$*+$2"-#"6$&%-.4$
! ,$'(7'8!$1%"!$7*#,$#,"$6$
! ?$'(7'8!$1%"!$7*#,$#,"$8$
! )$*!$#,"$/*!#'+0"$&-%.$#,"$&%0:!$#%$#,"$2"-#"6$%-$#,"$2"-#"6$#%$#,"$/*-"0#-*6$
! @A)$7,"+$#,"$)'-'9%('$%)"+!$:)$
! BA)$7,"+$#,"$)'-'9%('$%)"+!$/%7+$
C+"$'))(*0'#*%+$%&$'$)'-'9%('$*+2%(2"!$(*1,#*+1;$5,"$&%0:!$%&$'$)'-'9%('$,'!$'$
)-%)"-#8$7,*0,$*!$*.)%-#'+#$*!$0%+!#-:0#*+1$('.)!$'+/$(*1,#*+1;$$D+$
':#%.%9*("$,"'/(*1,#$*!$0%+!#-:0#"/$%&$'$-"&("0#*2"$!:-&'0"$7,*0,$*!$)'-'9%(*0;$
5,"$'0#:'($(*1,#$9:(9$*!$)('0"/$'#$#,"$&%0:!;$5,"$(*1,#$&-%.$#,*!$9:(9$-"&("0#!$
%&&$#,"$!:-&'0"$*+$)'-'(("($9"'.!>$7,*0,$E&%0:!F$#,"$,"'/(*1,#$9"'.$/*-"0#(8$
&%-7'-/$&-%.$#,"$0'-;$5,*!$*!$9"0':!"$'+8$(*1,#$!%:-0"$'#$#,"$&%0:!$%&$'$
)'-'9%('$-"&("0#!$*+$9"'.!$)'-'(("($#%$#,"$'6*!$%&$!8.."#-8$%&$#,"$)'-'9%(';$!
!
!"#$#%&'$()'*+,-$#.%'./'-'0-"-1.2-')%'$()'4.5,6'-%7'8#")5$"#9'
!"#"#$"%&'''
(x - h)2
= ±4 p(y - k) '''''! ()"%"'*)+,-'.'/"%0"1'
! 2'34'0)"'5340678"'9%:#'0)"'9:8;4'0:'0)"'/"%0"1+':%'0)"'/"%0"1'0:'0)"'53%"80%31'
! <'34'26%6$:=6':2"74';2+'>'39'0)"'26%6$:=6':2"74'5:(7'
'
*9-:;2)<''4.5,6'='>?@AB@''8#")5$"#9''C=DE'
?06%0'$@'5%6(37A'6'2380;%"&'
B%:#'0)"'2380;%"'30C4'"64@'0:'4""'0)60'0)"'26%6$:=6'#;40'
:2"7';2'*30'#;40'8;%/"'6(6@'9%:#'0)"'53%"80%31-+'4:';4"'
<D2E''F)"'/"%0"1'#;40'$"')6=9(6@'$"0(""7'0)"'9:8;4'*G+H-'
675'0)"'53%"80%31'*@.>I-''4:'0)"'/"%0"1'#;40'$"'*G+I-.*)+,-''
F)"'5340678"'9%:#'0)"'9:8;4'0:'0)"'/"%0"1'34'G'4:'2.GE''
F)"'"J;603:7':9'0)"'26%6$:=6'34&''''*1>G-G.K*@>I-'
!
!"#$)'$()')+,-$#.%'./'$()';-"-1.2-'F#$('$()')%'/.5,6'-%7'7#")5$"#9G'
IE''B:8;4'*L+K-+'M3%"80%31'@.>G' ' ' GE''B:8;4'*>I+'G-+'M3%"80%31'@.IN'
'
'
'
'
'
'
'
'
'
'
'
'
'
HE''B:8;4'*>H+'IG-+'M3%"80%31'@.D' ' DE''B:8;4'*O+>IN-+'M3%"80%31'@.G'
'
!
!
!
!
!
!
!
!
!
!
!"#"$%&"'()*+',%-./'"01'2)#3-*#)4'5%#6/+33*'
5#)*3'*+3'37."*)%0'%8'*+3'9"#"$%&"'()*+'*+3':);30'8%-./'"01'1)#3-*#)4<'
!"##$%&'(#)*+,-.,#/012&3104#56*+## # +"##$%&'(#)7,#!.,#/012&3104#568#
#
#
#
#
9"##$%&'(#):,#*9.,#/012&3104#56!## # 9"##$%&'(#)9,7.,#/012&3104#56+#
#
#
#
#
;"##$%&'(#)*+,*!.,#/012&3104#56*+# # -"##$%&'(#)*;,#!;.,#/012&3104#569#
#
#
#
#
<"##$%&'(#)*9,#*:.,#/012&3104#56;# # :"##$%&'(#)<,9.,#/012&3104#56;#
!
!
!
!
=>?>@AB'
5#)*3'*+3'37."*)%0'%8'*+3'9"#"$%&"<'
8"####=21324#)*9,7.,#$%&'(#)*9,:.#
!>"##$%&'(#)+,*-.,#?6+,#%?2@(#A%B@#
!!"##=21324#)9,*;.,#$%&'(#)9,*9.#
!+"##=21324#)#*+,#*!.#C@A#$%&'(#)#+,#*;."#
!9"##=21324#)#+,#9.,##/012&3104#5#6#;#
!7"##$%&'(#)#9,#>.,##?#6+#,#%?2@(#'?#
5#)*3'*+3'37."*)%0'%8'*+3'9"#"$%&"')0'*+3'8%#C'
(x - h)2
= ±4 p(y - k) ,#0"C3'*+3';3#*34D'
8%-./D'1)#3-*#)4D'"01'"4)/'%8'/ECC3*#E<'
!;"##4+#D#-4#D#!>5#D#!#6#>#
!-"##4+#E#:4#E#:5#E#9+#6>#
#
!
!"#"$%&
'()&*+)&,)-./.*.0/&0-&1&21314051&*0&,)3.6)&*+)&)781*.0/9&
!
!"#$"%&'"(%"()%*)+','"'-,.%
/'0"#,1)%+$-2%3-',"%45678%"-%+-190%4(6:8%"-%;%/'0"#,1)%+$-2%3-',"%45678%"-%*'$)1"$'5%47%;%:<38%
x - h( )2
+ (y - (k + p))2
= y - k - p( ) !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
Quadratic Modelling Word Problems
1. A ball is thrown upward into the air with an initial velocity of 80 ft/sec. The
formula h(t) = 80t – 16t2 give its height h(t) after t seconds.
a) What is the height of the ball after 2 seconds?
b) What is the maximum height of the ball?
c) How long does it take the ball to reach its maximum height?
d) How long is the ball in the air?
When an equation is NOT given:
1. Define your variable(s)
2. Write an equation(s) to solve the problem.
3. State the solution.
4. Explain in words how you found the solution.
1. The length of a rectangular pool is 4 yd longer than its width. The area of the
pool is 60 yd. What are the dimensions of the pool? (6 x 10 yds)
2. A rectangle has a perimeter of 52 inches. Find the dimensions of the rectangle
with maximum area. (13 x 13 in)
3. Find two consecutive negative integers whose product is 240. (-15 & -16)
4. Find two numbers who sum is 20 and whose product is a maximum (10 & 10).
!
!
!"#$%&#"'()*%!"#+,-)).%/0%
!"##$#%&'()#*&++,,-#./#0&'&12+'(3#.-',#'4(#&.)#/,#'4&'#.'/#4(.54'6#.-#7('()/6#&8'()#!#/(0,-3/#./#
7,3(+(3#*9#'4(#(:2&'.,-%4;'<#=#>?"@'A#B#AC'#B#A"?%
&< D,%#4.54#./#'4(#*&++,,-#&8'()#!#/(0,-3E#
#
*< F4&'#./#'4(#7&G.727#4(.54'#,8#'4(#*&++,,-E#
#
0< D,%#+,-5#3,(/#.'#'&H(#'4(#*&++,,-#',#)(&04#.'/#7&G.727#4(.54'E#
#
3< F4(-#%.++#'4(#*&++,,-#4.'#'4(#5),2-3#&-3#*2)/'E#%%
A"##I.-3#'%,#0,-/(02'.J(#1,/.'.J(#.-'(5()/#%4,/(#1),320'#./#?KA"##
#
#
L"##I.-3#'%,#-27*()/#%4,#3.88()(-0(#./#M#&-3#%4,/(#1),320'#./#.-.727"#
#
#
?"##$#)(0'&-5+(#4&/#().7('()#,8#?M#.-04(/"##I.-3#'4(#3.7(-/.,-/#,8#'4(#)(0'&-5+(#%.'4#
7&G.727#&)(&"#
#
#
#
N"##I.-3#'4(#-(5&'.J(#.-'(5()#%4,/(#/:2&)(#./#!O#7,)(#'4&-#L#'.7(/#'4(#.-'(5()"#
#
#
#
K"###P-(#/.3(#,8#&#)(0'&-52+&)#5&)3(-#./#A#93#+(//#'4&-#'4(#,'4()#/.3("##Q8#'4(#&)(&#,8#'4(#5&)3(-#./#
KL#93A6#8.-3#'4(#3.7(-/.,-/#,8#'4(#5&)3(-"#
#
#
#
C"##I.-3#'%,#-27*()/#%4,#/27#./#>!A#&-3#%4,/(#1),320'#./#&G.727"#
ANSWERS: 1) a. 24.5m b. 39.6m c. 2.8 sec d. 5.6 sec 2) 20 & 21 3) 4 & -4 4) 12 x 12 in 5) -2
6) 7 x 9 yds. 7) -6 & -6
!"#$%&#"'()*%!"#+,-)).%/0%
!"##$%#&#'()%*+&,-#./0.1*2)#&3(#,&/245(6#7*%5#,&38(#4&%&./,%)#&26#&*3#4&2292)"#:2#
92(#,&/245-#%5(#5(*85%#9'#&#./0.1*2#*2#'((%#&;9+(#%5(#839/26#&'%(3#!#)(4926)#*)#
096(,(6#;<#5=%>#?#@!A%B#C#!DD%#C#!B#
&> E*26#%5(#0&F*0/0#5(*85%#9'#%5(#./0.1*2"##
;> G5(2#6*6#%5(#./0.1*2#3(&45#*%)#0&F#5(*85%H##
4> G5(2#6*6#%5(#./0.1*2#5*%#%5(#839/26H##
6> G5&%#7&)#%5(#5(*85%#9'#%5(#./0.1*2#&'%(3#I#)(4926)H##
(> G5(2#7&)#%5(#./0.1*2#&%#(*85%#9'#!DD#'((%H#
B"##$2#9;J(4%#*)#%53972#/.7&36#*2%9#%5(#&*3#7*%5#&2#*2*%*&,#+(,94*%<#9'#!BK#'((%#.(3#
)(4926"#L5(#'930/,&###M=%>#?##!BK%#N#!A%B##8*+()#*%)#5(*85%#&'%(3#O%P#)(4926)"##
&>##G5&%#*)#%5(#5(*85%#&'%(3#B#)(4926)H##
;>##G5&%#*)#%5(#0&F*0/0#5(*85%#3(&45(6H##
4>##E93#597#0&2<#)(4926)#7*,,#%5(#9;J(4%#;(#*2#%5(#&*3H#
Q"##R/..9)(#<9/#&3(#%9))*28#&2#&..,(#/.#%9#&#'3*(26#92#&#%5*36@)%93<#;&,492<"##$'%(3###
%#)(4926)-#%5(#5(*85%#9'#%5(#&..,(#*2#'((%#*)#8*+(2#;<#5=%>#?#@!A%B#C#QK"S%#C#"TA"##
U9/3#'3*(26#4&%45()#%5(#&..,(#J/)%#&)#*%#3(&45()#*%)#5*85()%#.9*2%"##M97#,928#69()#
%5(#&..,(#%&1(#%9#3(&45#<9/3#'3*(26-#&26#&%#75&%#5(*85%#&;9+(#%5(#839/26#69()#
<9/3#'3*(26#4&%45#*%H#
S"##L5(#;&3;(3V)#.39'*%#=.>#(&45#7((1#6(.(26)#92#5*)#45&38(#=4>#.(3#5&*34/%"##W%#*)#
096(,(6#;<#.#?#@BDD4B#C#BSDD4#N#SXDD"##G5&%#.3*4(#)59/,6#5(#45&38(#'93#%5(#
,&38()%#.39'*%H##G5&%#*)#%5(#0&F*0/0#.39'*%H#
I"# L5(#.&%5#9'#&#;&)(;&,,#&'%(3#*%#5&)#;((2#5*%#*)#096(,(6#;<#%5(#'/24%*92###############
5=6>#?#@"DDQB6B#C#6#C#Q-#75(3(#5#*)#%5(#5(*85%#*2#'((%#9'#%5(#;&)(;&,,#&26#6#*)#%5(#
6*)%&24(#*2#'((%#%5(#;&)(;&,,#*)#'390#590(#.,&%("##G5&%#*)#%5(#0&F*0/0#5(*85%#
3(&45(6#;<#%5(#;&)(;&,,H##M97#'&3#*)#%5(#;&)(;&,,#'390#590(#.,&%(#75(2#*%#
3(&45()#*%)#0&F*0/0#5(*85%H#
A"###E*26#B#2/0;(3)#759)(#)/0#*)#QA#&26#759)(#.396/4%#*)#�&F*0/0"#
#
X"##E*26#B#2/0;(3)#759)(#6*''(3(24(#*)#SD#&26#759)(#.396/4%#*)#�*2*0/0"#
#
K"##$#3(4%&28,(#5&)#&#.(3*0(%(3#9'#SD#0(%(3)"##E*26#%5(#6*0(2)*92)#9'#%5(#3(4%&28,(#
7*%5#%5(#0&F*0/0#&3(&"#
#
T"##Y*41#5&)#!BD#'((%#9'#'(24*28#'93#(22(,"##W'#5*)#59/)(#*)#%9#;(#/)(6#&)#92(#)*6(#9'#
%5(#1(22(,-#'*26#%5(#6*0(2)*92)#%9#0&F*0*Z(#%5(#&3(&"#
%
!"#$%&#"'()*%!"#+,-)).%/0%1%234356%
!"# $#%&'(()*#+,&&-).#%'/&-.%.#%0&'/10#%0)#0-.%'&-+#2-.%&-+%#'3#4'.%'5#.)&6-51#788#
+/.%'9)&.#,#2,*"##:0)#+0,&1)#-.#;<#=)&#=)&.'5"##:0)#+'9=,5*#>'/(2#('.)#?8#
=,..)51)&.#,#2,*#3'&#),+0#;!#-5+&),.)#-5#3,&)"##@0,%#+0,&1)#>'/(2#A)#%0)#9'.%#
=&'3-%,A()B#,52#>0,%#-.#%0)#9,C-9/9#=&'3-%"#
#
#
?"##$#+('%0-51#.%'&)#.)((.#D8#=,-&.#'3#E),5.#2,-(*#,%#;78#),+0"##:0)#'>5)-1/&).#%0,%#
3'&#),+0#;7#-5+&),.)#-5#=&-+)B####?#3)>)&#=,-&.#>-((#A)#.'(2#),+0#2,*"##@0,%#=&-+)#
.0'/(2#A)#+0,&1)2#%'#9,C-9-F)#=&'3-%G#
#
#
7"##$#.H/,&)B#>0-+0#-.#?#-5"#A*#?#-5"B#-.#+/%#3&'9#),+0#+'&5)&#'3#,#&)+%,51/(,&#=-)+)#'3#
9)%,("##:0)#.-2).#,&)#3'(2)2#/=#%'#9,I)#,#A'C"##J3#%0)#A'%%'9#9/.%#0,6)#,#
=)&-9)%)&#'3#7?#-5"B#>0,%#>'/(2#A)#%0)#()51%0#,52#>-2%0#3'	,C-9/9#6'(/9)G##
@0,%#-.#%0)#9,C-9/9#6'(/9)G#
#
#
D"# #K/21-51#A*#0-.#=,.%#=)&3'&9,5+)#'5#9,%0#)C,9.B#L'..#+,5#).%-9,%)#%0)#1&,2)#0)#
>-((#&)+)-6)#'5#,#9,%0#)C,9#/.-51#%0)#3/5+%-'5##*#M#N%?#O<%#O#P<B##>0)&)#%#
&)=&).)5%.#%0)#5/9A)&#'3#0'/&.#.=)5%#.%/2*-51"##@0,%#-.#%0)#()7,.#5/9A)&#'3#
0'/&.#0)#0,.#%'#.%/2*#-5#'&2)&#%'#&)+)-6)#,#1&,2)#'3#Q8G#
#
R"##S-52#,#5/9A)&#>0'.)#.H/,&)#-.#RR#1&),%)&#%0,5#T#%-9).#%0)#5/9A)&"#
#
T"# #J&9,IU.#J%)9.#9,I).#+/.%'9#0,52N=,-5%)2#:N.0-&%."##:0)#+'9=,5*U.#=&'E)+%)2#
,55/,(#&)6)5/)#+,5#A)#9'2)()2#A*#%0)#3/5+%-'5#LVCW#M##!?C?##O#!?8C##O!!!B#>0)&)#
C#-.#%0)#5/9A)&#'3#-%)9.#=&'2/+)2#89%-:9$#)$,"##:0)#+'.%#%'#=&'2/+)#%0).)#-%)9.#
+,5#A)#9'2)()2#A*#%0)#3/5+%-'5#XVCW#M#DC?#O#7RC#O#Q!"##J3#%0)#+'9=,5*#.)((.#)6)&*#
-%)9#%0,%#-%#=&'2/+).B#-%#>-((#9,I)#,#=&'3-%#'3#;???"##Y'>#9,5*#-%)9.#2-2#%0)#
+'9=,5*#.)((G#
#
P"# #$#&)+%,51()#-.#T#+9#('51#R#+9#>-2)"##@0)5#),+0#2-9)5.-'5#-.#-5+&),.)2#A*#%0)#
.,9)#5/9A)&#%0)#,&),#-.#%&-=()2"##S-52#%0)#5/9A)&#),+0#2-9)5.-'5#>,.#
-5+&),.)2#A*"#
#
#
!"# #$#%&'()*+,-)%#./&-0#1/(2#)%&)#344456#/7#&*'-87&0#9:#;44#5#8.#.&*'/*+"##</*0#(2&#
0/5&*7/8*#8.#(2&#./&-0"#
#
="# #$#%&'()*+,-)%#)*/5)-#>&*#1/(2#)%&)#?64456#2)7#8*/0&#)-8*+#)#9)%*"##@2(2&%#
(2%&/0&7#)%&#&*'-87&0#9:#?44#5#8.#.&*'/*+"##</*0#(2�/5&*7/8*7#8.#(2&#>&*"#
#
#
?4"##$#>87/(/A&#*,59&%#/7#8*:%&#(2)*#/(7#%&'/>%8')-"##</*0#(2&#*,59&%"#
#
??"##@18#>87/(/A&#*,59&%7#2)A&#)#7,5#8.#3#)*0#)#>%80,'(#8.#3"##</*0#(2&#*,59&%7"#
#
?6"##$#3/*#9:#B/*#>28(8+%)>2#/7#7,%%8,*0&0#9:#)#.%)5.#,*/.8%5#1/0(2"##@2&#)%&)#
8.#(2&#.%)5&#&C,)-7#(2&#)%&)#8.#(2&#>28(8+%)>2"##</*0#(2/0(2#8.#(2&#.%)5&"#
#
#
?;"# $#1)-D1):#8.#,*/.8%5#1/0(2#2)7#)%&)#B6#56#)*0#7,%%8,*07#)#71/55/*+#>88-#
(2)(#/7#!#5#1/0&#)*0#?4#5#-8*+"##</*0#(2/0(2#8.#(2)-D1):"#
#
#
?E"##@2&#(8()-#7,%.)'&#)%&)#8.#(2&#%&'()*+,-)%#78-/0#7281*#/7#;F#56"##</*0#(2&#A)-,.#G"#
#
#
#
#
?3"##$#98G#1/(2#2&/+2(#HG#I#3J#'5#2)7#)#7C,)%	)7/(2#7/0&#G#'5"##$#7&'8*0#98G#
1/(2#2&/+2(#HG#I#6J#'5#2)7#)#7C,)%	)7/(2#7/0&#HG#I#?J#'5"##K.#(2#98G&7#
2)A&#(2)5&#A8-,5&L#./*0#G"#
#
#
?F"##$#98G#1/(2#)#7C,)%	)7&#)*0#*8#-/0#/7#(8#9)0&#.%85#)#7C,)%&#>/&'.#5&()-#
9:#',((/*+#7C,)%&7#.%85#(2&#'8%*&%7#)*0#.8-0/*+#,>#(2/0&7"##@2&#',(M8..#7C,)%&7#
)%#'5#8*#)#7/0&"##K.#(2&#A8-,5.#(2bG#/7#?44#'5;L#./*0#(2�/5&*7/8*7#8.#
(2%/+/*)-#>/&'.#5&()-"#
x
2x
x +2
Quadratic Equations Review #1
Solve each equation by factoring.
1. x2 ! 4x ! 32 = 0 2. 4x
2 + 20x = 0 3. d
2 ! 29d = -100 4. 18x
2 + 29x + 3 = 0
Solve each equation by completing the square.
5. x2 + 4 = 8x 6. x
2 !
5x = 8 7. 2x
2 ! 12x = 8 8. 4x
2 !12x = 16
Solve each equation by using the quadratic formula.
9. x2 + 2x = 7 10. 2x
2 ! 12x + 5 = 0 11. 2x ! 5x
2 + 3 = 0 12. 6x
2 ! 3x + 2 = 0
Solve each equation by using any method.
13. 3x2 + 6x +3 = 0 14. x
2 + 6x = 4 15. 2x
2 + x !1 = 3 16. 3x
2 + 2 = -7x
17. r2 = 3r + 70
18. (x ! 3)
2 = 6 19. 6x
2 – 8x + 9 = 4 20. 4x
2 + 8x = -3
Write the equation of the parabola with the given info:
21. Focus (-1, 6) Directrix y=0 22. Focus (3,-2) Directrix y=-4
Given a = -3 + 2i and b= 4-5i
23. Find a+ b 24. Find a – b 25. Find the product of a and b
26. Find 2a – 3b 27. Find a2 – b
2
Use your calculator to answer the following questions:
28. A ball is thrown upward vertically with an initial speed of 96 feet per second. The
equation h = 96t – 16t2 gives the height of the ball in t in seconds. What is the
maximum height reached by the ball? When will the ball be 128 feet above its
starting point?
29. Terry has 200 yards of fencing to enclose a rectangular garden on three sides. The
fourth side will be the side of the house. What dimensions of the garden will
maximize the area?
(HONORS) Write the quadratic equation with the given solutions
30. 3, -8 31. -5, 32.
33. 34. 35.
(HONORS) Solve by factoring.
36. x4 ! 6x
2 + 5 = 0 37. a
3 ! 81a = 0 38. 39.
!"#$%#&'()*%+,-+.*/0! ! ! ! !
1-22*-3*45+*627389*:;9+*<745+<74-=72*+>?@+99-A39B+C;74-A39D*9E<6A29D*A@*.A@F9GH*
! )4FH*1A@<I!! ! J+@4+>*1A@<I!! !
"#!!$%&!'%!(%)!*+,'!-./!0/1-/2!
!!!!!!&+-.%)-!3!4354)53-%16!
!
!
!
7#!$%&!'%!(%)!*+,'!-./!32+8!%*!!
!!!!!8(99/-1(6!
! !
:#!!;+0/,!-./!<313=%53>!*+,'!-./!!
!!!!!+,'+43-/'!035)/8?!!
*
!@!!!!!!!!!!!!!!!!"@!!!!!!!!!!!!!!!#@!!!!!!!!
!@!!!!!!!!!!!!!!!!$@!!!!!!!!!!!!!!!%@!!!!!!!
A#!!$%&!'%!(%)!4.3,B/!3!<313=%53!*1%9!0/1-/2!-%!8-3,'31'!*%196!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
D#!!E.3-!+8!-./!<1%4/88!4355/'!*%1!4.3,B+,B!3!<313=%53!*1%9!8-3,'31'!-%!0/1-/2!*%196!CCCCCCCCCCCCCCCCCCCCC!
F#!!$%&!'%!(%)!-/55!+*!3!<313=%53!%</,8!)<!%1!'%&,6!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
G#!!E.3-!+8!-./!<3--/1,!*%1!B13<.+,B!-./!<31/,-!(!@!276!!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
H#!!E./,!(%)!B13<.!3!<313=%53!.%&!'%!(%)!I,%&!+*!-./!<3--/1,!+8!'+**/1/,-!*1%9!-./!<31/,-6!!;+0/!3,!/239<5/>!
3,'!8-3-/!-./!,/&!<3--/1,#!!
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
J#!!E.3-!'%/8!+-!9/3,!-%!K8%50/L!3!M)3'13-+46!!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
"N#!!E.3-!&3(8!'%!(%)!I,%&!-%!8%50/!3!M)3'13-+46!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
""#!!E.3-!+8!-./!8-13-/B(!*%1!8%50+,B!3!M)3'13-+4!&./,!355!-./!2O8!31/!+,!3!8M)31/6!!CCCCCCCCCCCCCCCCCCCCCCCC!
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
""#!!E1+-/!-./!M)3'13-+4!*%19)53?!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
":#!!!!&'@!CCCCCCCCCC! "A#!&('@!CCCCCCCCCC!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"D#!!P%1!3,(!<%8+-+0/!1/35!,)9=/1!!>! !CCCCCCCCCCCCCCC!
"F#!!!E./,!43,!(%)!5/30/!3,!&(!+,!(%)1!3,8&/16!!!CCCCCCCCCCCCCCCCCCCCCCCCCC!
"G#!!E.3-!'%!.>!I>!3,'!<!1/<1/8/,-!+,!-./!/M)3-+%,!%*!3!<313=%53!&1+--/,!+,!-./!*%19?!
(x - h)2
= ±4 p(y - k) !
.@!CCCCCCCCCCCCCCC!!!I!@!CCCCCCCCCCCCCCCC!!<!@CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
$%&!'%!(%)!I,%&!&./,!-%!)8/!QA<!3,'!&./,!-%!)8/!RA<6!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
"H#!!S./!<3--/1,!*%1!*34-%1+,B! !+8!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!
S./!<3--/1,!*%1!9)5-+<5(+,B!T !!+8!CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC!!!!!!!!!!
:KLML%)G!"J#!!P%1!3!M)3'13-+4!/M)3-+%,!+,!8-3,'31'!*%19!T!!!!!!!!!!!!!!!!!!!!!!!!!U>!!-./!8)9!%*!-./!1%%-8!@!CCCCCCCCCC!
3,'!-./!<1%')4-!%*!-./!1%%-8!@!CCCCCCCCCCCCC#!
!"#$%&'()!$#*&+*,+&-"%#
!"#$$%&'($)*+$,+-)+.$/'($/.&0$12$0344+)-3$21-$)*+$5/-/617/$$$$$$$$$$$$$$$$$$$$$$$$$$$$$888888888888888888888888888888$
!9#$$%&'($)*+$,+-)+.$/'($/.&0$12$0344+)-3$21-$)*+$5/-/617/$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$88888888888888888888888888888$
!!#$$:*/';+$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$)1$0)/'(/-($21-4$8888888888888888888888888888888888888888888888888888$
!<#$$:*/';+$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$)1$,+-)+.$21-4$888888888888888888888888888888888888888888888888888888$
!=#$$>1+0$)*+$5/-/617/$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$15+'$?5$1-$(1@'A$888888888888888888888888888888888888888888$
!B#$$C2)+-$31?$571)$)*+$,+-)+.D$@*/)$&0$)*+$5/))+-'$21-$;-/5*&';$$$$$$$$$$$$$$$$$$$$$$$$$$$$888888888888888888888888888$
!E#$$F17,+$63$2/G)1-&';$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$88888888888888888888888888888888888888888888888888888888888$
!H#$$F17,+$63$?0&';$)*+$I?/(-/)&G$21-4?7/$$$$$$$$$$$$$$$$$$$$$$$$$$$$$88888888888888888888888888888888888888888888$
!J#$$F17,+$63$G1457+)&';$)*+$0I?/-+$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$88888888888888888888888888888888888888888888888$
!K#$$L*/)$&0$)*+$+I?/)&1'$12$/$5/-/617/$@&)*$%1G?0$M9DNBO$$/'($>&-+G)-&.$3P<$
888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888$
<"#$$F&457&23$ $$$888888888888888888$$$$<9#$$F&457&23$ $88888888888888888$
.("!"%/0$<!#$$%&'($)*+$I?/(-/)&G$+I?/)&1'$@&)*$-11)0$$!$/'($N<$$88888888888888888888888888888888888888$
.("!"%/0$<<#$$%&'($)*+$I?/(-/)&G$+I?/)&1'$@&)*$-11)0$ $$$$$88888888888888888888888888888888888888$
<=#$$%/G)1-$ $$888888888888888888888888888888888888888888888888888888888888888888888888$
$%&'()!$#*&+*,+&-"%#
<B#$$F17,+$63$;-/5*&';$1'$31?-$G/7G?7/)1-$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$88888888888888888888888888888888888888888$
<E#$$9<#$C$6/77$&0$)*-1@'$?5@/-($,+-)&G/773$@&)*$/'$&'&)&/7$05++($12$9!J$2++)$5+-$0+G1'(#$Q*+$+I?/)&1'$$
$$$$$$$*$P$9!J)$R$9E)!$;&,+0$)*+$*+&;*)$12$)*+$6/77$@*+-+$)$&0$)*+$'?46+-$12$0+G1'(0$/2)+-$)*+$6/77$&0$-+7+/0+(#$
L*/)$&0$)*+$6/77S0$4/.&4?4$*+&;*)A$8888888888888$$$T1@$71';$&0$)*+$6/77$&'$)*+$/&-A8888888888888$ $
<H#$$%&'($)@1$G1'0+G?)&,+$510&)&,+$&')+;+-0$@*1$5-1(?G)$&0$K"#$
>+2&'+$,/-&/67+M0OU$$ $ $ $ $ L-&)+$/'$+I?/)&1'M0OU$
F)/)+$)*+$017?)&1'U$88888888888888888888888$
V.57/&'$&'$@1-(0$*1@$31?$?0+($)*+$+I?/)&1'$)1$2&'($)*+$017?)&1'U$8888888888888888888888888888888888888$
$
888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888$
<J#$$%&'($)@1$'?46+-0$@*10+$0?4$&0$!"$/'($@*10+$5-1(?G)$&0$/$4/.&4?4#$
>+2&'+$,/-&/67+M0OU$$ $ $ $ $ L-&)+$/'$+I?/)&1'M0OU$
F)/)+$)*+$017?)&1'U$88888888888888888888888$
V.57/&'$&'$@1-(0$*1@$31?$?0+($)*+$+I?/)&1'$)1$2&'($)*+$017?)&1'U$8888888888888888888888888888888888888$
$
888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888$
!"#$%&"'()*+',*)&-./01'2#%3&+#%4516',*0)1&'76'894%':6';&11*9'<'
!
!"#$%&'(#)%"*#%*+%,-"%.)/,*012$344)%"4*/"&*5'"(#)%"4*!
"#$$%&'!(%#!)%*+!,%*!-!.%/$-0(!12-1!3'&450&6!7#483&6!-03!1'&1&!*43'&!,%*!
-/#&'/'01&!$-*+&9!!:%#*!1'-/!4&!40!.2-*5'!%,!3'&450405!-!8%05!*%88'*!.%-&1'*9!!;0'!
/%*04056!(%#*!1'-/!4&!2-03'3!&+'1.2'&!12-1!&2%)!43'-&!,%*!1)%!&'.14%0&!%,!12'!0')!
*%88'*!.%-&1'*9!!!
!
!!
:%#*!1-&+!4&!1%!,403!-85'7*-4.!,#0.14%0&!)412!5*-$2&!12-1!/-1.2!12'!1)%!&+'1.2'&9!!
<2'!,#0.14%0&!)488!7'!#&',#8!40!.2'.+405!&-,'1(!,'-1#*'&!%,!12'!3'&4506!84+'!'&14/-1'3!
$''3!-03!2'4521!-!=-*4%#&!$%401&!%,!12'!1*-.+9!!<2'(!)488!-8&%!7'!'&&'014-8!40!$8-00405!
/-0#,-.1#*'!%,!12'!.%-&1'*!1*-.+!-03!&#$$%*1!,*-/'9!!!
!
67)"8*9:%'#*67)4*;)#'/#)%"<*
-9!!>2-1!,-/484-*!,#0.14%0&!2-='!5*-$2&!12-1!/-1.2!-88!%*!$-*1&!%,!12'!3'&450!
&+'1.2'&?!
!
!
!
!
79!!!>2-1!&1*-1'54'&!.%#83!(%#!#&'!1%!,403!,#0.14%0&!)412!5*-$2&!12-1!/%3'8!12'!
&+'1.2'&?!
!
!
!
!
.9!!>2-1!3%!(%#!1240+!-*'!12'!+'(!$%401&!%0!'-.2!&+'1.2!12-1!&2%#83!7'!#&'3!40!
,403405!-!,#0.14%0!/%3'8!,%*!12'!5*-$2!$-11'*0?!!
!
!
!
!
"#$!%&''$%!(&)*+$%!,$*-./!*0$+(#$*!)%$!/&+!1&,$'$,!2$''!34!.%)5#*!&6!'-/$)%7!
$85&/$/+-)'7!&%!9:),%)+-(!6:/(+-&/*;!!<:+!+#$%$!)%$!&+#$%!0-/,*!&6!)'.$3%)-(!6:/(+-&/*!
+#)+!,&!#)=$!*:(#!.%)5#*;!!!
>/!+#-*!'$**&/!4&:!2-''!3$.-/!*+:,4!&6!)/!-15&%+)/+!(')**!&6!)'.$3%)-(!6:/(+-&/*!()''$,!
!"#$%"&'(#)*+%,-'"%.;!!?&:!2-''!'$)%/!#&2!+&!:*$!5&'4/&1-)'!6:/(+-&/!+&!1&,$'!
(&15'$8!.%)5#-()'!5)++$%/*7!'-0$!+#$!%&''$%!(&)*+$%!,$*-./;!!
<$6&%$!4&:!)+!+#$!%&''$%!(&)*+$%!,$*-./*!-/!1&%$!,$+)-'7!1)0$!*:%$!4&:!0/&2!#&2!+&!
,&!(:%=$!6-++-/.!-/!4&:%!()'(:')+&%;!!!"#$*$!)%$!+#$!*+$5*@!
AB!C/+$%!8!=)':$*!-/!DA!)/,!4!=)':$*!-/!DE!F!"#"$%$&'("B!
EB!G#&&*$!+#$!+45$!&6!(:%=$!4&:!2)/+!F!"#"$%$)#*+B!
! ,-$.(/012$3#4$5$67$$$$FD-/$)%!H$.%$**-&/B!
$ 8-$9:#'012!!!!!!!!!!!!!!!!FI:),%)+-(!H$.%$**-&/B!
$ ;-$&4<012$$$$ !!!!!!!!!!FC85&/$/+-)'!H$.%$**-&/B!
JB!K:+!+#$!%$*:'+-/.!%$.%$**-&/!$9:)+-&/!-/!?A!+&!*$$!+#$!.%)5#@!
$$$$$$$$$$$=>$$%$$?@0!$%$8-!"#"(A"(+A$$%$&9$%$B-012&9$ $
"#$!6&''&2-/.!,-).%)1!*#&2*!+#$!.%)5#!&6!)!6:/(+-&/!+#)+!(&:',!%$5%$*$/+!)!%&''$%!
(&)*+$%!,$*-./!3)*$,!&/!+#$!L$(+-&/!A!*0$+(#;!
!
! ! ! ! ! ! ! ! !
A;!!M&:',!'-/$)%7!9:),%)+-(7!&%!
$85&/$/+-)'!6:/(+-&/*!5%&=-,$!.&&,!
1&,$'*!6&%!+#-*!.%)5#N!!F>6!/$$,$,!
+%4!+#$1!)/,!*$$B;!
!
!
!
!
!
!
E;! ?&:%!()'(:')+&%!#)*!),,-+-&/)'!1&,$'-/.!&5+-&/*!*:(#!)*!C-):6(+012!!6&%!)!(:3-(!
6:/(+-&/;!
)B! M#)+!0$4!5&-/+*!&/!+#$!.%)5#7!3$*-,$*!FA7OB7!,&!4&:!+#-/0!2&:',!3$!#$'56:'!-/!
6-/,-/.!)!(:3-(!6:/(+-&/!+#)+!1&,$'*!+#$!5%&5&*$,!L$(+-&/!A!,$*-./N!!
!
!
3B!!P*-/.!+#$!5&-/+*!4&:!*$'$(+$,!-/!5)%+!)B7!:*$!C-):6(+012!+&!6-/,!)!(:3-(!6:/(+-&/!
+&!1&,$'!+#$!L$(+-&/!A!,$*-./;!!!
C9:)+-&/!&6!4&:%!(:3-(!6:/(+-&/@!!
!"## $%&'(#)*+%#!+,-!#.+/!0-*/#1*234#.%*1#,"5##673#8+%3#0*#830#)*+%#!&4!+4&0*%98#
:-/2*:#0*#1&0!(#0(3#;3!0-*/#<#238-=/5"##>*1'&%3#)*+%#=%&'(#0*#0(3#8(&'3#*.#
0(3#;3!0-*/#<#238-=/5##?38!%-,3#:&)8#0(&0#)*+%#!+,-!#.+/!0-*/#-8#*%#-8#/*0#&#
=**2#1*2345#
#
#
@5##A(&0#2*#)*+#0(-/B#:*+42#,3#0(3#1-/-1+1#/+1,3%#*.#'*-/08#/33232#0*#.-/2#&#
!+,-!#1*234#.*%#&0&#*%#=%&'(#'&003%/C###6D380#)*+%#-23&8#*/#0(3#!&4!+4&0*%5"#
#
E5##D(3%3#&%3#8*13#8-=/-.-!&/0#2-..3%3/!38#,30:33/#0(3#238-=/#-23&8#.*%#;3!0-*/#<#&/2#
;3!0-*/#F#*.#0(3#'%*'*832#%*443%#!*&803%5##D(3#/3G0#2-&=%&1#8(*:8#&#=%&'(#*.#&#
.+/!0-*/#0(&0#!*+42#%3'%383/0#&#%*443%#!*&803%#238-=/#,&832#*/#0(3#;3!0-*/#F#
8B30!(5#
#
&"# H-/2#0(3#!**%2-/&038#*.#B3)#'*-/08#*+04-/-/=#0(3#8(&'3#*.#0(-8#=%&'(5##I8-/=#
0(383#'*-/08#.-/2#&/2#=%&'(#&#!+,-!#.+/!0-*/#1*234#&/2#!*1'&%3#-0#0*#0(3#
=%&'(#*.#0(3#'%*'*832#;3!0-*/#F#238-=/5###J8#)*+%#1*234#&#=**2#.-0#0*#0(3#
238-=/C#
#
,"## K*+%#!&4!+4&0*%#(&8#80-44#1*%3#1*234-/=#*'0-*/8#8+!(#&8#!"#$%&'()*##.*%#&#
L+&%0-!#.+/!0-*/5##A(&0#2*#)*+#0(-/B#1-=(0#,3#0(3#1-/-1+1#/+1,3%#*.#'*-/08#
/33232#0*#.-/2#&#L+&%0-!#1*234#.*%#&0&#*%#=%&'(#'&003%/C#
#
!"###H-/2#&#L+&%0-!#.+/!0-*/#1*234#.*%#0(3#;3!0-*/#F#238-=/5##
ML+&0-*/#*.#)*+%#L+&%0-!#.+/!0-*/N##
!"## $%&'(#)*+%#L+&%0-!#.+/!0-*/#1*234#.*%#0(3#;3!0-*/#F#238-=/5##>*1'&%3#)*+%#
=%&'(#0*#0(3#8(&'3#*.#0(3#;3!0-*/#<#238-=/5##?38!%-,3#:&)8#0(&0#)*+%#L+&%0-!#
.+/!0-*/#-8#*%#-8#/*0#&#=**2#1*2345#
#
#
!"#$%"&'(#)*
!"#$%&'($&)*(&+,$-.#+/(+$)/0(&*$&0+$1#&+*&./2$3#02./"0-4$$5%(-($3#02./"0-$&*($3*"6$&$
)&*7(*$2)&--$2&))(+$!"#$%"&'(#)*+%,-'"%.4$$8#9/2$&0+$1#&*./2$3#02./"0-$&*($&)-"$
:"),0"6/&)$3#02./"0-4$$$
;$&"%"&'(#*/-$&0$(<:*(--/"0$.%&.$/-$(/.%(*$&$*(&)$0#69(*=$&$'&*/&9)(=$"*$.%($:*"+#2.$"3$
*(&)$0#69(*-$&0+$'&*/&9)(-4$$$;$!"#$%"&'(#$/-$&$6"0"6/&)$"*$.%($-#6$"3$6"0"6/&)-$$
;$!"#$%"&'(#)*+%,-'"%$/-$&0,$3#02./"0$>/.%$&$*#)($.%&.$2&0$9($>*/..(0$/0$.%($3"*6?$
!"#$%&%%'(#(%)%'(*+#(*+%)%,%%)%'+#%)%'-%%%%%%>%(*($$&0$@$A=$$2"(33/2/(0.-$B&0=$C=$&AD$&*($*(&)$0#69(*-=$&0+$(<:"0(0.-$&*($0"00(7&./'($/0.(7(*-4$
;0,$&)7(9*&/2$(<:*(--/"0$/0$.%($3"*6$'(#(%)%'(*+#(*+%)%,%%)%'+#%)%'-%%/-$2&))(+$&$!"#$%"&'(#)/0!1/..'"%4$$$$
E0($"3$.%($6"-.$/6:"*.&0.$2%&*&2.(*/-./2-$"3$&0,$:"),0"6/&)$3#02./"0$"*$(<:*(--/"0$
/-$/.-$+(7*((4$$5%($2/31//)"*)()!"#$%"&'(#$/-$.%($7*(&.(-.$(<:"0(0.$"3$.%($'&*/&9)(4$$
F"*$(<&6:)(=$&$1#&+*&./2$:"),0"6/&)$%&-$+(7*(($G=$&$2#9/2$:"),0"6/&)$%&-$+(7*(($H=$
&0+$&$1#&*./2$:"),0"6/&)$%&-$+(7*(($I4$$;$0"0J(*"$2"0-.&0.$/-$&$:"),0"6/&)$"3$
+(7*(($A$B(<4$,KLD4$$5%(*#/(2'%3),"/**','/%-$"3$&$:"),0"6/&)$/-$.%($2"(33/2/(0.$"3$.%($
.(*6$>/.%$.%($%/7%(-.$+(7*(($
M0*.-(%2(12)*"1&*.%($.(*6-$"3$&$:"),0"6/&)$&*($>*/..(0$/0$+(-2(0+/07$"*+(*$9,$
+(7*((4$$N"$.>"$.(*6-$%&'($.%($-&6($+(7*(($B-/02($)/O($.(*6-$%&'($9((0$2"69/0(+D4$
P"),0"6/&)-$&*($,#(..'*'/2)4$?$
! .%($0#69(*$"3$.(*6-$BQK6"0"6/&)=$GK9/0"6/&)=$HK.*/0"6/&)=$IR$K:"),0"6/&)D$
! .%($+(7*(($$$
5%(-($&*($:"),0"6/&)-$/0$"0($'&*/&9)($/0$-.&0+&*+$3"*6?$$ $ $ $ $
Q4$$$S<I$R$H<G$R$I<$T$U$$$$B:"),0"6/&)=$+(7*(($ID$$$$$$$$$$$G4$$<$R$U$$B9/0"6/&)=$+(7*(($QD$ $
H4$$$TL$ $$$B6"0"6/&)=$+(7*(($AD$ $ $$$$$$$$$$$$$$$$$I4$$<L$T$H<H$T$V$$B.*/0"6/&)=$+(7*(($LD$
$ $ $ $$$$
5%(-($&*(*%+,$:"),0"6/&)-$/0$"0($'&*/&9)(?$
Q4$$W<H,L$R$G<G,S$T$I$$$$BG$'&*/&9)(-D$ $ $$$$$$$$$G4$$<$XH$$R$I<G$T$Q$B0(7&./'($(<:"0(0.D$ $
H4$$
5x7
+ 3x4
+2
x$$B'&*/&9)($/0$+(0"6/0&."*$6(&0-$&$0(7&./'($(<:"0(0.D$
$
-./00123*,45*2+..+6178*9+.37+:1/.0*;3*<58=55*>*7?:;5=*+2*,5=:0@*
Q4$$$3B<D$K$V<L$R$H<$T$QA$$$$$$$$$$$$$$$ $ $
G4$$$7B<D$K$G<V$R$H<I$R$<G$X$L<$T$QA$$$$$$
H4$$$%B<D$K$XL<H$X$I<$$
I4$$$OB<D$K$G<$
L4$$$6B<D$K$$H<L$R$U<S$X$QQ<G$X$L<H$R$V<$X$Q$$$$$$$$$$
GRAPHS of POLYNOMIAL FUNCTIONS:
!"#$%&#%'()#*%+"#' ' ''''''''',+#-&.'()#*%+"#' ' '''''''''''/)&0.&%+*'()#*%+"#'
'''1'2'3' ' ' ' ''''''''''''''1'2'45'6'7' ' ' '1'2'53'8'35'6'4'
9-:.--';' ' ' ' 9-:.--'<' ' ' 9-:.--'3'
' ' ' ' ' ' ' ' '
'
!)=+*'()#*%+"#' ' ' ''''''/)&.%+*'()#*%+"#' ' ' /)+#%+*'()#*%+"#'
>'2'54'6'75'8'3' ' ' ''''''''1'2'5
?'6'45
4'6'35
3'8'@5'8'<' ' ''''1'2'5
7'6'75
4'8'?5'
9-:.--'4' ' ' ' 9-:.--'?' ' ' ''''9-:.--'7'
END BEHAVIOR: Is the behavior of the graph as x approaches +! or -!
If the degree is EVEN, both ends have the SAME behavior
! If the leading coefficient is positive, both ends are up
! If the leading coefficient is negative, both ends are down
If the degree is odd, the ends have OPPOSITE behavior
! If the leading coefficient is positive, the right end is up, left down
! If the leading coefficient is negative, the right end is down, left up
Leading
Coefficient Degree Example x " - ! x " !
+ even f(x) = x2 f(x) " ! f(x) " !
- even f(x) = -x2 f(x) " -! f(x) " -!
+ odd f(x) = x3
f(x) " -! f(x) " !
- odd f(x) = -x3
f(x) " ! f(x) " -!
The Fundamental Theorem of Algebra says that every polynomial with degree greater
then zero has at least one complex root. An extension of this theorem says that:
A polynomial of degree n has exactly n complex roots
In other words …. the degree of a polynomial = # of zeros/roots/solutions
Ex. x3 + 4x2 + 4x = 0 has 3 zeros Ex. x4 – 10x2 + 9 = 0 has 4 solutions
! solutions, zeros, and roots are the values of x which give y = 0
! complex roots means real and/or imaginary
o complex numbers have the form a + bi
! ‘n’ counts multiple roots the number of times they occur
o multiplicity is the number of times a zero occurs
! imaginary roots always come in conjugate pairs (a + bi, a – bi)
! each x-intercept represents a real root of the polynomial equation
! a polynomial function with odd degree must have at least 1 real root
o the graph must cross the x-axis at least once (think about the end behavior)
! a polynomial function with even degree will have either no real roots or an even
number of real roots
o the graph may or may not cross the x-axis, but if it does it will cross an even
number of times (think about the end behavior)
! every polynomial of degree n > 0 can be written as the product of a constant k
and n linear factors. P(x) = k(x – r1)(x – r2) (x – r3) ….(x – rn)
! to find zeros write the polynomial in factored form and set each factor = 0
! for polynomial P(x), if a is a zero then P(a) = 0
When finding the zeros of polynomials REMEMBER:
! #zeros = degree of polynomial = # of factors
! if a is a zero then (x-a) is a factor
! when you divide a polynomial by one of it’s factors the remainder is 0
! you can use division to break a polynomial down into its factors (just like you do
with numbers)
! for quadratics you have multiple tools for finding the zeros (factor, complete
the square, quadratic formula, graphing)
Graphing Polynomial Functions Worksheet
To graph a polynomial function:
a. Find the zeros of the function. Remember real zeros = x-
intercepts so graph these points on the x-axis.
b. Find the y-intercept (value of y when x=0)
c. Determine the end behavior of the function based on the
degree and the leading coefficient.
d. Using the end behavior and the intercepts to make a smooth
curve.
Graph each function. USE GRAPH PAPER!
1.
y = -2 x2
- 9( ) x + 4( ) 2. y = (x2 -4)(x+3)
3. y = -1(x2-9)(x2-4) 4.
y =1
4x + 2( ) x -1( )
2
5.
y =1
5x - 3( )
2
x +1( )2
6.
y = x +1( )3
x - 4( )
7. y = x(x-1)(x+5) 8. y = x2 (x + 4) (x-3)
!"#$%"&'(#)*"%+),'-'.'"%))
/0)12))))!"#"$%$&#
'$($)*#$%$"+$$,$$"#$($)!! ! /0)32)))))))!'#
"$%$-#
'$($)&+$$,$$!'#$%$&+!$
)
"#$!%&'!"#()'*!+(,! ! ! ! ! ! "#$!%&'!"#()'*!+(,-./0*+1&%!2!3lizabeth -
2011 All Rights Reserved
)
456).768.)'%)756)89":6..)";)#"%+)<'-'.'"%)=6962)
45!!!6+7+$'!89+*(%!%'*:!+#%.!9+*(%!%'*:5!
! ;5!!<=>%+/>0!8=('!">>!%'*:(5!
! ?5!!!@=A%*"B%!
! C5!!D*+#1!6.)#!
! E5!!F3G3HI!
!"#$%&$'()*'+','-#!"#!$!#%&'()*(!+,(%&-!.&'!-"/"-"01!$!2&340&+"$3!54!$!3"0,$'!.$)(&'!
&.!(%,!.&'+!!"#$#%&6!!7(!)$0!5,!*#,-!"0!23$),!&.!(%,!#($0-$'-!3&01!-"/"#"&0!$31&'"(%+6!!)
89$+23,:!!!!!"#$%&%'#
'%(%)#
"%*%'#%*%+,%-%!#(",####
;33!(%,!/$'"$53,#!$0-!(%,"'!,92&0,0(#!$',!',+&/,-<!3,$/"01!&034!$!3"#(!&.!(%,!
)&,..")",0(#:!!=<!>?<!>@<!?<!A6!!B.-$&/!".!$!2&C,'!&.!"#"#!+"##"01!.'&+!(%,!2&340&+"$3<!$!
(,'+!C"(%!(%$(!2&C,'!$0-!$!D,'&!)&,..")",0(!+*#(!5,!"0#,'(,-!"0(&!(%,!)&'',)(!2&#"("&0!
"0!(%,!2&340&+"$36E!!F%,#,!0*+5,'#!.&'+!(%,!-"/"-,0-6!!!!
G,!.&'+!(%,!-"/"#&'!.&'!(%,!#40(%,(")!-"/"#"&0!*#"01!&034!(%,!)&0#($0(!(,'+!B=E!&.!(%,!
3"0,$'!.$)(&'!!"'(&6!BH&(,:!7.!(%,!-"/"#&'!C,',!!")(&<!C,!C&*3-!*#,!$!-"/"#&'!&.!>=6E!!
F%,!0*+5,'#!',2',#,0("01!(%,!-"/"#&'!$0-!(%,!
-"/"-,0-!$',!23$),-!"0(&!$!-"/"#"&0>3"I,!
)&0."1*'$("&06!!!J"'#(<!01'#2)*-3#!(%,!K=K!(%$(!
"0-")$(,#!(%,!3,$-"01!)&,..")",0(:!
456$'76"!54!(%,!0*+5,'!&0!(%,!3,.(<!$0-!)$''4!
(%,!',#*3(!"0(&!(%,!0,9(!)&3*+0:!
!!
8**!-&C0!(%,!)&3*+0:!
!
456$'76"!54!(%,!0*+5,'!&0!(%,!3,.(<!$0-!)$''4!
(%,!',#*3(!"0(&!(%,!0,9(!)&3*+0:!
!
!
8**!-&C0!(%,!)&3*+0:!
!
456$'76"!54!(%,!0*+5,'!&0!(%,!3,.(<!$0-!)$''4!
(%,!',#*3(!"0(&!(%,!0,9(!)&3*+0:!
!
!
8**!-&C0!(%,!)&3*+0:!
!"#$%&#'!"#!$%&!'()"&*!+'!$%&!,&-$.!/'0!1/**#!
$%&!*&2(,$!3'$+!$%&!'&4$!1+,()'5!
!
!!
())*0+6'!$%&!1+,()'!-+*!$%&!*&)/3'0&*5!
!
!!
7%&!1+)8,&$&0!039323+'!325!
!
!
:,,!'()"&*2!&41&8$!$%&!,/2$!"&1+)&!$%&!1+&--313&'$2!+-!$%&!;(+$3&'$!8+,#'+)3/,<!=3'1&!
6&!2$/*$&0!63$%!/!>$%!0&?*&&!8+,#'+)3/,!/'0!03930&0!3$!"#!/!,3'&/*!$&*).!$%&!;(+$3&'$!32!
/!@*0!0&?*&&!8+,#'+)3/,<!!7%&!,/2$!&'$*#!3'!$%&!*&2(,$!,32$!ABC!32!$%&!*&)/3'0&*<!7%&!
;(+$3&'$!/'0!*&)/3'0&*!1/'!"&!1+)"3'&0!3'$+!+'&!&48*&223+'5!!
B4@!D!4
B!E!@4!F!@!D!BGA4FBC!
AH+$&!$%/$!'+!039323+'!+8&*/$3+'2!6&*&!8&*-+*)&0!$+!1+)8($&!$%&!/'26&*!$+!$%32!
039323+'!8*+",&)<C!!!
+,-*.$-&.*%/*$,-*&012-..*13*.'/$,-$%2*)%4%.%1/*5-0-6*
! ! IC!!J*3'?!K+6'!
! ! BC!!L(,$38,#!
! ! @C!!:00!
! ! >C!!MNON:7!-*+)!BC!
!
!
!
!
!
!"#$%&$'()*'+','-#).%&#)$%&)(-&//'('&#$)-/)0)'#)$%&)1'+',-2)3)4) ) 567879!:)
Divide:
Step 1: To set up the problem, first, set the denominator
equal to zero to find the number to put in the division
box. Next, make sure the numerator is written in
descending order and if any terms are missing you must
use a zero to fill in the missing term, finally list only the
coefficient in the division problem.
Step 2: Once the problem is set up correctly, bring the
leading coefficient (first number) straight down.
Step 3: Multiply the number in the division box with
the number you brought down and put the result in the
next column.
Step 4: Add the two numbers together and write the
result in the bottom of the row.
Step 5: Repeat steps 3 and 4 until you reach the end of
the problem.
Step 6 : Write the final answer. The final answer is
made up of the numbers in the bottom row with the last
number being the remainder and the remainder must be
written as a fraction. The variables or x’s start off one
power less than the original denominator and go down
one with each term.
Multiply everything by 1/2 to
eliminate the fraction in the
denominator
2x^2 - 3x - 2 + 3/(2x - 1)
is the answer
!
Dividing Polynomials - EXAMPLES
!
"#$#%#&'!()!*!+,&,+#*-!
./!012342)!5!.64
6)6!7!.84
6)9!:!01;4
6)9! ! !
!
!"#"$%&'(")*&+,)*&!"#"(",)&
6/!0;46!7!4!7!<9!:!024!5!.9! ! !!!!!!!!!!!!!!!!2/!! !0=4
6!7!64!5!;9064!7!29
1.! !
! !
!
!
!
!
!
=/!0=42!7!84
6!5!24!7!89!:!064!7!.9!!!!!!!!!!!!!!!!!>/!!064
2!7!24
6!7!.84!7!89!:!04!7!=9!
!
!
!
!
!
;/!!
(2x4
+ 3x3
+ 5x -1) ! (x2
- 2x + 2)!
!
!!
!
!
!
! ! !!
!"#"$%&'(")*&-.)/0%/"1&!"#"(",)&
</!0646!5!24!7!=9!:!04!7!69!!!!!!!!!!!!!!!!!!!!!!!!!8/!!04
=!7!24
2!5!>4!7!;9!:!04!5!69!
!
!
!
!
?/!0642!5!=4!7!;9!:!04!5!29!!!!!!!!!!!!!!!!!!!!!!!!!.3/!04
=!7!64
2!5!;4
6!784!5!.39!:!04!5!69!
!
&
../&0=@=!7!>@
6!5!6@!5!29!A!06@!7!.9!
Polynomial Division Worksheet
Divide using Synthetic Division
1. (3y3 + 2y
2 – 32y + 2) / (y – 3)
2. (2b3 + b
2 – 2b + 3) / (b + 1)
3. (2c3 – 3c
2 + 3c – 4) / (c – 2)
4. (3x3 – 2x
2 + 2x – 1) / (x – 1)
5. (t4 – 2t
3 + t
2 – 3t + 2) / (t – 2)
6. (3r4 – 6r
3 – 2r
2 + r – 6) / (r + 1)
7. (z4 – 3z
3 – z
2 – 11z – 4) / (z – 4)
8. (2b3 – 11b
2 + 12b + 9) / (b – 3)
9. (6s3 – 19s
2 + s + 6) / (s – 3)
10. (x3 + 2x
2 – 5x – 6) / (x – 2)
11. (x3 + 3x
2 – 7x + 1) / (x – 1)
12. (n4 – 8n
3 + 54n + 105) / (n – 5)
13. (2x4 – 5x
3 + 2x – 3) / (x – 1)
14. (z5 – 6z
3 + 4x
2 – 3) / (z – 2)
15. (y4 + 3y
3 + y – 1) / (y + 3)
Divide using long division:
16. (4s4 – 5s
2 + 2s + 3) / (2s – 1)
17. (2x3 – 3x
2 – 8x + 4) / (2x + 1)
18. (4x4 – 5x
2 – 8x – 10) / (2x – 3)
19. (6j3 – 28j
2 + 19j + 3) / (3j – 2)
20. (y5 – 3y
2 – 20) / (y – 2)
!"#$"%&'"(%"()%*+,-.+/'#,%!0"1%2%!!"#"$!"#"%%%#.3%3'4'3)%5-%"()%,'.)#$%")$/%&!"#"'(%
%%%%%%6+.7%3'4'8'+.9%%%%%%%%%% % % % !-."()"':%
;'4'8'+.9%%
%
%
%
%
%
%
%
%%%%%%%%%%<(=89%
%%%&!!"#"$!"#"%(%>%&!"#"'(%%2%0!)"*"'!"*"+( + !">0?@A1"
B=,"'*,-%5+"(%8'3)8%5-%0?@A1%"+%7)"9%%%%%
!!"#"$!"#"%%%2%%%&!"#"'("&!
)"*"'!"*"+(%%C%%!,
!'.:)%!0"1%2%!!"#"$!"#"%-%
% % "&!(%2%&!"#"'("&!)"*"'!"*"+(%%C%%!,"
D8'.7%#"&!(%2%&!"#"'("&!)"*"'!"*"+(%%C%%!,""""
E(#"%'8%"()%4#=)%+F%!0#1G%HHHHHHHH%
"
Remainder Theorem: The value of the polynomial p(x) at x=a is the same
as the remainder you get when you divide that polynomial p(x) by x – a.
! To evaluate a polynomial p(x) at x = a, use synthetic division to divide
the polynomial by x = a. The remainder is p(a).
Use the Remainder Theorem and synthetic division to find f(4) where
f(x) =
The Remainder Theorem tells us that if we use synthetic division and divide
f(x) by (x-4), the remainder will be equal to f(4).
The remainder is 127. So f(4) = 127.
Factor Theorem: p(a) = 0 if and only if x – a is a factor of p(x).
! If you divide a polynomial by x = a and get a zero remainder, then, not
only is x = a a zero of the polynomial, but x – a is also a factor of the
polynomial.
Determine whether x + 4 is a factor of each polynomial.
Note: synthetic division can be used instead of long division !
!"#$%&!'())*+,-.)/012,-304)5)627894):;09401+)
!"##$%#&'("#)#*(+#,#(
-#,#+(#.#!/0##%123#&'45"#67128#792:;<:1=#76>7:1:6:1?2@#
#45# *# 4!# /# /# 4+# !/#
#
#
# # # # # # # # # &'45"#
A?:<B#&C688128#12#,5#%?D#(#E1CC#81F<#:;<#7GH<#D<76C:I#
5"## G@##$%#%'("#)#5(-#,#J(
5#.#+(#,#K0#%123#%'*"#
!"##$%#&''"#)#L'*#,#+'
5#.#-'#,#!K#%123#M'*"##
=@##$%#&''"#)#*'-#.#5'
*#.#-'#%123#M'4+"#
#*"##N<:<DH12<#E;<:;<D#'#)#5#17#G#O<D?#?%##&''"#)#*'K#,#'
-#.#5'
*#,#+'
5#,#-#
P?D#'#)#5#:?#><#G#O<D?#?%##&''"0###&#'5"#H67:#<FGC6G:<#:?#QQQQQ@##
#
-"##N<:<DH12<#E;<:;<D#'#)#,-#17#G#7?C6:1?2#?%###'L#.#+'
+#.#+'
-#.#+'
*#.#5'
5#,#!/'#,#J#)#/#
$%##'#)#,-#17#G#7?C6:1?20#E;G:#H67:#><#G#%G=:?D#?%#:;<#D<CG:<3#%62=:1?2R#QQQQQQQQQQ#
#
+"##$7#(#,#5#G#%G=:?D#?%#%'("#)#(-#,#-(
*#.#+(
5#.#-(#,#!5R#
#5# !# 4-# +# -# 4!5#
#
#
#
4ST4####P123#%'5"0#1%#1:#<U6GC7#O<D?0#:;<2#(#,#5#17#G#%G=:?DI#
L"##PG=:?D#%'("#)#5(*#.#!!(
5#.#!J(#.#V#81F<2#:;G:#%'4*"#)#/@##'H<G27#:;G:#(#.#*#17#G#%G=:?D"#
#
#
K"##PG=:?D#%'("#)#*(*#.#!-(
5#,#5J(#,#5-#81F<2#:;G:#(#,#5#17#G#%G=:?D@#
#
$%#:;<#D<HG123<D#)#/0#
:;<2#'(45"#17#G#%G=:?D@#
#
!"##$%&'(#)*+)#,#%-#+#.'/0#01#123"#4#35#6#,3
,#6#73#8#9!######:%(;#+<<#0)*'/#.'/0-=#
,# 9# >,# >7# 9!#
# # #,# #?# >9!#
#
# 9# ?# >7# ####?# # # 3,#6#7#4#?#
######################################################### # # # #3,#4#7#
# # # # # # # 3#4#5@#>5#
7"##A0<&'#B%&'(#)*+)# %-#+#/00)#01#12C"#4#,C5#6#DC
,#6#95C#6#D=#
#
#
9?"##A0<&'#E23"#4#35#8#3#8#9?@#%1##>,#%-#+#/00)=#
#
#
#
!"#$#%&'()*+,(-./0./1(2,+*34((
"5$64##F-'#)*'#-GCHE/0;GI)#/G<'#)0#J/%)'#+#KG+;/+)%I#1+I)0/@#)*'(#G-'#<0(B#;%&%-%0(#)0#;%&%;'#)*'#B%&'(#'KG+)%0(#
LM#)*'#KG+;/+)%I#1+I)0/#M0G#10G(;=##A')#)*'#KG0)%'()#4#?#+(;#1%(;#)*'#/'C+%(%(B#/00)-=#
99="##A0<&'#3N#6#O3
5#8#O3
,#8#,N3#6#N?#4#?##B%&'(##5#8#%#%-#+#/00)=#
#
#
#
#
9,="##A0<&'#3N#8#3
5#8#O3
,#6#9N3#6#,?#4#?##%1##69#8#5%#%-#+#/00)=#
Practice 6-3 Dividing Polynomials
Determine whether each binomial is a factor of x3 ± 3x2 – 10x – 24.
1. x + 4 2. x - 3 3. x + 6 4. x + 2
Divide using synthetic division.
5. (x3 - 8x2 + 17x - 10) ! (x - 5) 6. (x3 + 5x2 - x - 9) ! (x + 2)
7. (-2x3 + 15x2 - 22x - 15) ! (x - 3) 8. (x3 + 7x2 + 15x + 9) ! (x + 1)
9. (x3 + 2x2 + 5x + 12) ! (x + 3) 10. (x3 - 5x2 - 7x + 25) ! (x - 5)
11. (x4 - x3 + x2 - x + 1) ! (x - 1) 12.
13. (x4 - 5x3 + 5x2 + 7x - 12) ! (x - 4) 14. (2x4 + 23x3 + 60x2 - 125x - 500) ! (x + 4)
Use synthetic division and the Remainder Theorem to find P(a).
15. P(x) = 3x3 - 4x2 - 5x + 1; a = 2 16. P(x) = x3 + 7x2 + 12x - 3; a = -5
17. P(x) = x3 + 6x2 + 10x + 3; a = -3 18. P(x) = 2x4 - 9x3 + 7x2 - 5x + 11; a = 4
Divide using long division. Check your answers.
19. (x2 - 13x - 48) ! (x + 3) 20. (2x2 + x - 7) ! (x - 5)
21. (x3 + 5x2 - 3x - 1) ! (x - 1) 22. (3x3 - x2 - 7x + 6) ! (x + 2)
Use synthetic division and the given factor to completely factor eachpolynomial function.
23. y = x3 + 3x2 - 13x - 15; (x + 5) 24. y = x3 - 3x2 - 10x + 24; (x - 2)
Divide.
25. (6x3 + 2x2 - 11x + 12) ! (3x + 4) 26. (x4 + 2x3 + x - 3) ! (x - 1)
27. (2x4 + 3x3 - 4x2 + x + 1) ! (2x - 1) 28. (x5 - 1) ! (x - 1)
29. (x4 - 3x2 - 10) ! (x - 2) 30.
31. A box is to be mailed. The volume in cubic inches of the box can be expressed as the product of its three dimensions:V(x) = x3 - 16x2 + 79x - 120. The length is x - 8. Find linearexpressions for the other dimensions. Assume that the width is greater than the height.
(3x3 2 2x2 1 2x 1 1) 4 ax 1 13b
ax4 1 53x
3 2 23x
2 1 6x 2 2b 4 ax 2 13b
Name Class Date
Lesson 6-3 Practice Algebra 2 Chapter 64
© P
ears
on E
duca
tion,
Inc.
All
right
s re
serv
ed.
A2_3eCSF06_002-029 8/5/02 6:48 PM Page 4
!"#$%&'&(")*+,-"#&./"$#")0&1*23$#&./"$#")&4$#5%/""3&
6789&1+,-&3/"&:"#$%&$;&3/"&<$=>,$)+*=%&*,-&%3*3"&3/"&)?=3+<=+2+3>&$;&"*2/&:"#$@&
!"# $%&'#(#%#&#)#*#'+#%#+&#,#*#'#
-"####$%&'#(#-&.#,#/&
*#,#!0&
+#
+"# $%&'#(#%#1&-#,#-.'
*##%#&
-#)#!2'#
*"###$%&'#(#%&-#)#&#,#-'
-#%&
-#,#*'#
."## $%&'#(#&#%&#)#-'+#%&#,#.'#
2"###$%&'#(#-&#%&)+'-%*&#,#!'#
6A'B&&4#+3"&*&<$=>,$)+*=&"C?*3+$,&+,&%3*,-*#-&;#$)&/*D+,E&3/"&E+D",&#$$3%@&
3"###-4#+45#!#
/"###!#6789:;8:<:9=#-4#0#
1"###5-#6789:;8:<:9=#-4#!4#-#
67F87G&H%"&%>,3/"3+2&-+D+%+$,&3$&%/$I&3/*3&2&+%&*&:"#$&$;&;JKLM&&
!0"#$%&'#(#+&*#)#/&
+#,#-&
-#,#!0&#)#*#>##<#(#5-#
!!"##$%&'#(#*&+#,#1&
-#,#/&#,#+#>##<#(#+#
!-"##$%&'#(#-&+#)#.&
-#,#*&#,#+#>##<#(#!#
!+"#$%&'#(#-&*#)#&
+#,#!*&
-#)#.&#)#2#>##<#(#5+#
!*"#$%&'(##*&+#,#2&
-#)#/&#,#+>#<#(#?##
!."##$%&'#(#-3&*#,#1&
+#)#+&
-#)#2&#)#!#>##<#(#5!@+#
!"#$$%&'()*$+,-.$/$0-1$2$"-
3$4$1-$5+$6)7$89):$,-2!.$5;$&$+&'()*#$
!<#$%&'()*$+,-.$/$-1$43-
3$4$0-$2$!=$5+$6)7$89):$,-21.$5;$&$+&'()*#$
!=#$$%&'()*$6$/$-1$4$>-
3$4$1-$2!=$5+$6)7$89):$(?&($,-23.$5;$&$+&'()*#$$
!0#$$$@?):$(?&($41$5;$&$AB*)$)+$C7D(5ED5'5(6$3$)+$(?B$E)D69)C5&D$+79'(5)9$$$
F,-./$->$2$<-
1$2$!1-
3$4$1-$4!=$&9G$B-E*B;;$F$,-.$&;$&$E*)G7'($)+$D59B&*$+&'()*;#$
3H#$@?):$(?&($4!$5;$&$AB*)$)+$$C7D(5ED5'5(6$>$)+$(?B$E)D69)C5&D$+79'(5)9$$$$$$$$$$$$$$$$$$$$$$$$$$$
$+,-./$-I$2$-
>$4$"-
1$4$!>-
3$4$!!-$41$&9G$B-E*B;;$+$,-.$&;$&$E*)G7'($)+$D59B&*$+&'()*;#$
3!#$$%59G$&$E)D69)C5&D$+79'(5)9$)+$GBJ*BB$>$;7'?$(?&($K)(?$43$&9G$1$&*B$AB*);$)+$$C7D(5ED5'5(6$3#$$
33#$$%59G$&$E)D69)C5&D$+79'(5)9$)+$GBJ*BB$I$;7'?$(?&($43$5;$&$AB*)$)+$C7D(5ED5'5(6$1$&9G$>$5;$&$AB*)$)+$
C7D(5ED5'5(6$3#$
31#$$LB(B*C59B$8$;)$(?&($(?&($$+,-$.$/$-1$2$8-
34$8-$2!H$$5;$G5M5;5KDB$K6$-$21#$
3>#$$%59G$8$;)$(?&($:?B9$-1$4$-
3$48-$2$!H$5;$G5M5GBG$-$41$N$(?B$*BC&59GB*$5;$43#$
3I#$$%59G$8$;)$(?&($:?B9$-1$$4$8-
34$8-$2!$5;$G5M5GBG$K6$-O3N$(?B$*BC&59GB*$/H$
3"#$$LB(B*C59B$8$;)$(?&($(?&($$+,-$.$/$-1$2$8-
34$8-$2!H$$5;$G5M5;5KDB$K6$-$21#$
3<#$$@PQRS$$$-1$2$>-
3$4$I-$$/$H$')CEDB(BD6#$
!"#$%&'()*)+,'
3=#$$@PQRS$$$->$2$<-
3$4!=$/$H$')CEDB(BD6#$$
30#$LB(B*C59B$&DD$M&D7B;$)+$8$;)$(?&($$+,-$.$/$83-3$4$>8-$21$$5;$G5M5;5KDB$K6$-$4$!#$
10#$$%59G$(?B$*BC&59GB*$5+$(?B$E)D69)C5&D$1-!HH
$2$I-=I$4$>-
1=$23-
!<$4$$"$5;$G5M5GBG$K6$-2!$
1!#$$T*5(B$&$'7K5'$BU7&(5)9$?&M59J$AB*);$3N$ $&9G$O!#$
13#$$T*5(B$(?B$U7&*(5'$BU7&(5)9$?&M59J$AB*);$$3!$$&9G$$1$4$!$#$
11#$$T*5(B$(?B$'7K5'$BU7&(5)9$?&M59J$AB*);$$ $$$&9G$3$2$1!$
1>#$$@PQRS$$$->$2$-
1$2$"-
3$4$!>-$4$3H$/$H$J5MB9$(?&($1$4$!$$5;$&$*))(#$$
1I#$$@PQRS$$$->$2$-
1$2$"-
3$4$!>-$4$3H$/$H$89):59J$$O!$4$1!$"$5;$&$*))(#$
! ! ! ! ! ! ! ! ! ! ! ! "#$#%&!
'()*+,(-!./,-),0123!!"#$!%!&#'()#*+%'!,-)./+#)!!0"12!+,!#!%)3!$!%$4!/5#!)-*64$7!7-.8!/8%/!!%#&!+7!
)49%/+:4!064'#5!/84!;<%;+71!%)3!!%$&!+7!+/+:4!0%6#:4!/84!;<%;+712!/84!,-)./+#)!8%7!%/!'4%7/!=!$4%'!
>4$#!64/544)!#!%)3!$?!
! !@,!!%#&!%)3!!%$&!8%:4!#&+/4!7+9)72!/84)!!0"1!8%7!%!>4$#!64/544)!!!!!!!#!%)3!$'!
%*+,(-*1!%((+!452(/263!!@,!!
p
q !+7!+)!7+*&'47/!,#$*!%)3!+7!%!$%/+#)%'!$##/!#,!%!&#'()#*+%'2!/84)! !+7!
%!,%./#$!#,!/84!.#)7/%)/!0'%7/1!/4$*!%)3! !+7!%!,%./#$!#,!/84!'4%3+)9!.#4,,+.+4)/?!
! @,!%!&#'()#*+%'!8%7!$%/+#)%'!$##/72!/84)!/8#74!$##/7!5+''!64!,$%./+#)7!#,!/84!,#$*!A!
0,%./#$!#,!/84!.#)7/%)/!/4$*1!B!A!0,%./#$!#,!/84!'4%3+)9!.#4,,+.+4)/1?!!!
!"#$%&'())*+,-)#..)/01)%())*+&')2#/+3,#.)233/4)35)))60"1!C!!D"!E!F!"!
G!<=H"!
D!<!H"!F!H!!
I/4&!=J!!"%./#$7!#,!.#)7/%)/!/4$*J!A=!2!AG!2!AH!!0/84!&K71!
I/4&!DJ!"%./#$7!#,!'4%3+)9!.#4,,+.+4)/J!A=!2!AD!!!0/84!LK71!
I/4&!GJ!M#77+6'4!:%'-47!#,! J!A !2!A !2!A !2!A !2!A !2!A !?!!
N8474!.%)!64!7+*&'+,+43!/#J!A=!2!A !2!AG!2!A !2!AH!2!A !?!
N#!,+)3!/84!*)+7*1!$##/7!#,!/84!4L-%/+#)2!/47/!/84!M+6'4!$##/7!-7+)9!7()/84/+.!
3+:+7+#)!0(#-K$4!'##O+)9!,#$!%!P!$4*%+)34$1?!!Q).4!(#-!,+)3!D!$##/7!(#-K''!8%:4!
%!L-%3$%/+.!4L-%/+#)!/8%/!.%)!64!7#':43!5+/8#-/!9-477+)9?!!
&899:%;3!
! N#!4:%'-%/4!%!&#'()#*+%'!!0"1!%/!"!C!#2!-74!7()/84/+.!3+:+7+#)!/#!3+:+34!/84!&#'()#*+%'!
6(!"!C!#'!!N84!$4*%+)34$!+7!!0#1?!
! @,!(#-!3+:+34!%!&#'()#*+%'!6(!"!C!#!%)3!94/!%!>4$#!$4*%+)34$2!/84)2!)#/!#)'(!+7!"!C!
#!!%!>4$#!#,!/84!&#'()#*+%'2!6-/!0"!R!#&!+7!%'7#!%!,%./#$!#,!/84!&#'()#*+%'?!
! @,!!%#&!%)3!!%$&!8%:4!#&+/4!7+9)72!/84)!/84$4!+7!%!>4$#!64/544)!#!%)3!$'!
! @,!%!&#'()#*+%'!8%7!$%/+#)%'!$##/72!/84)!/8#74!$##/7!5+''!64!,$%./+#)7!#,!/84!,#$*!A!
0,%./#$!#,!/84!.#)7/%)/!/4$*1!B!A!0,%./#$!#,!/84!'4%3+)9!.#4,,+.+4)/1?!!!
! %!&#'()#*+%'!,-)./+#)!5+/8!#33!349$44!*-7/!8%:4!%/!'4%7/!=!$4%'!$##/!!
! %!&#'()#*+%'!,-)./+#)!5+/8!4:4)!349$44!5+''!8%:4!4+/84$!)#!$4%'!$##/7!#$!%)!4:4)!
)-*64$!!
! %!$##/!.%)!8%:4!%!*-'/+&'+.+/(!S!=!058+.8!*4%)7!+,!(#-!,+)3!%!$%/+#)%'!$##/!-7+)9!7()/84/+.!
7-67/%/+#)2!(#-!*+98/!5%)/!/#!/47/!+/!%9%+)1!
!
HONORS
EXAMPLES: Find the possible rational roots, then find all the zeros.
1.
3x3
- x2
-15x + 5 = 0 2.
x4
- 5x3
+ 9x2
- 7x + 2 = 0
PRACTICE:
1. Solve
2. Solve
3. Find all the zeros of
f x( ) = x4
- x3
+ 2x2
- 4x - 8
4. Find all the roots of
5. Find all the zeros of
6. Find all the solutions of
0 = 15x4
+ 68x3
- 7x2
+ 24x - 4
Rational Root Theorem Worksheet
Find all the roots.
1.
p x( ) = x4
+ 5x3
+ 5x2
- 5x - 6 2.
p x( ) = x3
- 5x2
- 4x + 20
3.
p x( ) = x4
- 5x3
+ 9x2
- 7x + 2 4.
p x( ) = x3
- 2x2
- 8x
5.
p x( ) = x3
+ 7x2
+ 7x -15 6.
p x( ) = 2x3
- 5x2
- 28x +15
7.
p x( ) = x3
- 7x - 6 8.
p x( ) = x4
+ 2x3
- 9x2
- 2x + 8
!"#$%"&'(#)*+,'+-)./)
./)0)1)23+456)46+)78(96:);%")78(96'%7)5(#5<#(4"8=)
!"#!$%&#'#$(!)*&$%#+#,&$%#+#*&$%#(#-&-######## # -"###!$%&#'#$(!).&$%#/#,&-$%#/#!&$%#+#-&-###
##################################################################################################
# # # # # # # # #
#
#
#
#
#
#
#
#
#
#
#
#
#
#
Graphing Calculator Allowed
3. P(x) = $0%*#+#,%1#(#-%#+#2&##(##$*%-#(##-%#+#2&##
3�#9:8;<_________________________________________________
b) Degree ___________________ c) Classify by the # of terms: __________________________# # #
__________________4. Find p(3) for p(x) = -4x4 + 9x
3 + 10x
2 – 2x + 17 #
#
========================*"##Is (x + 2) a factor of p(x) = x4 + 3x
3 – 3x – 10 ?
#
."###>?@?7A#BC?6D##"%7)>',':'"%)$-x4 – 5x
3 + 7x
2 + 2x + 4) ÷ (2x – 3) )))
___________________________________ )
0"##Write a polynomial function in standard form that has zeros at 2, -1, and 3 multiplicity 2?
___________________________________
E"##4:F@A#G#'##%,#(#%-##(#!!%#+#,##D?@A6#5H35#/,#?C#3#IA8:"###
___________________________________
;?"%"8:=#1"##Write a polynomial function in standard form that has zeros at 2, -1, and 1 ± 3i ?
___________________________________
;?"%"8:=#!G"##Find all the zeros of ?
___________________________________
a.) >AD8AA================##
b.) %/#?65A8JAK5C================##
c.) L/?65A8JAK5#=====================#
#
%#
L#
d.) >AD8AA================##
e.) %/#?65A8JAK5C================##
f.) L/?65A8JAK5#=====================#
#
%#
L#
Polynomial Review #2
!"##$%#&'#(#)*+,#,-#./01#2##304#5#60'#7#80#5#!9#:##;<=#,+#><=#?,@:#
3"##$%#/0#5#81#(#-(A@,+#,-#./01#2#04#5#60'#5#!B03#5#B0#7#!4#:##;<=#,+#><=#?,@:#
'"#CD?E#./'1#-,+#./01#2#'04#7#!!0'#7#03#5#!B0#7#!3##:#
4"##C(A@,+##./01#2#'0'#5#!403#7#80#7#!9##A,F.G*@*G=H##IDJ*?#./&B1#2#9###
B"##;+D@*#@<*#.,G=?,FD(G#D?#-(A@,+*E#-,+F#>D@<#)*+,%K##!#[email protected]@=#'H##9H#&4#:#
M"##N,GJ*#./01#2##0'#7#'03#7#!!0#7#8##IDJ*?#@<(@#&!#D%#(#)*+,"#####
8"##C(A@,+#./01#2#M0'#7#3'03#7#M0#5#O##D-#/0#7#41#D%#(#-(A@,+"#
O"##N,GJ*#P8"#####
6"#!NQ*@A<#@<*#I+(.<#,-!!"/01#2#&!/0#7#31/0#5#'1/0#5#!1###/?,#A(GA1#
!9"#N,GJ*#./01#2#0'#7#!!03#5#'M0#7#'M##D-#/0#7#M1#D%#(#-(A@,+"#
!!"##N,GJ*#./01#2#!B0'#7#!!603#7#!90#5#!M##D-#O#D%#(#)*+,"#
!3"##RDJDE*#04#S#'0'#5#!O03#S!30#5#!M#T=#0#7#'#L%D?I#G,?I#EDJD%D,?"#
!'"#U?*#+,,@#,-#30'#S!903#5#60#S4#2#9#D%#4"##CD?E#@<*#,@<*+#+,,@%"#
!4"#$-#'#5#3##D%#(#)*+,#,-#(#.,G=?,FD(GH#><(@#<(%#@,#T*#(?,@<*+#)*+,:#
!B"#R*%A+DT*#@<*#*?E#T*<(JD,+#,-#*(A<K##/(1##$/01#2#0B#S#0'#S#03#5#0#5#3V# /T1##%/01#2#S04#S#603#
!M"#W..+,0DF(@*#@,#@<*#?*(+*%@#@*?@<#@<*#+*(G#)*+,%#,-#$/01#2#0'#SM03#5#O0#S3"##/X%*#(#A(GALG(@,+1#
!8"#C,+#=#2#0/0#5#'1/0#S#!13H#E*@*+FD?*#@<*#)*+,%#(?E#@<*D+#[email protected]@="#
!O"#;+D@*#(#.,G=?,FD(G#-L?A@D,?#>D@<#)*+,%#!#(?E#3#/,-#[email protected]@=#'1#D?#%@(?E(+E#-,+F"#
!6"#X%*#%=?@<*@DA#EDJD%D,?#@,#-D?E#$/S31#D-#$/01#2#40B#5#!904#S#!!0'#S3303#5#390#5#!9"#
39"#C(A@,+K##30'#5#!B03#S!40#S4O##D-#/0#S#31#D%#(#-(A@,+"#
3!"#R*@*+FD?*#D-#@<*#E*I+**#,-#@<*#-L?A@D,?%#T*G,>#D%#*J*?#,+#,EE"##Y,>#F(?=#+*(G#)*+,%#E,*%#*(A<#<(J*:#
#
(1# # # ######T1# # # # # A1# # #
# # #
#
#
#
#
#
"#$%$&'(#33"##CD?E#(#@<D+E#E*I+**#.,G=?,FD(G#>D@<#)*+,%#&4#(?E#3#7#'D"#
"#$%$&'(#3'"#;+D@*#(#ALTDA#*ZL(@D,?#D?#')*%+*&+!,$&-#<(JD?I#)*+,%#'#(?E#3#5##"#
"#$%$&'(#34"#CD?E#(#.,G=?,FD(G#*ZL(@D,?#<(JD?I#+,,@%#&3#(?E#'#5##"#
"#$%$&'(#3B"#CD?E#(GG#)*+,%#-,+#"/01#2#304#5#'0'#5#M03#5#!30#&#O#D-#3##D%#(#)*+,"#/?,#A(GA1#
"#$%$&'(!3M"##CD?E#(GG#@<*#[UNN$\]^#+(@D,?(G#+,,@%#,-#./01#2#'04#5#!90'#&#O03#5#0#&#!B#
"#$%$&'(#3M"#CD?E#(GG#@<*#+,,@%#-,+K#-/01#2#'04#5#!40'#5#!403#&#O0#7#O##/?,#A(GA1#
"#$%$&'(#38"#CD?E#(GG#@<*#+,,@%#-,+#./01#2#'0'##&#M0#5#3#2#9##/?,#A(GA1#
