9.1 adding and subtracting polynomials. monomial is an expression that is a number, variable, or a...
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9.1 Adding and Subtracting Polynomials
MonomialIs an expression that is a number, variable, or a product of a number and one or more variables.
Ex: 3, y, -4x2y, c/3
c/3 is a monomial, but c/x is not because there is a variable in the denominator.
Degree of a Monomial
Is the sum of the exponents of its variables. For nonzero constant the degree is zero.
Ex 1: Find the degree of the monomial.
a)18 degree ___
The degree of a nonzero constant is __.
Ex 1: Find the degree of the monomial.
a)18 degree _0_
The degree of a nonzero constant is _0_.
b) 3xy3 Degree ___
The exponenta are ___ and ___. Their sum is ___
b) 3xy3 Degree _4_
The exponents are _1_ and _3_. Their sum is _4_
c) 6c Degree: __ 6c = 6c1. The exponent is ___.
c) 6c Degree: _1_ 6c = 6c1. The exponent is _1_.
PolynomialIs a monomial or the sum or difference of two or more monomials.
Ex: 4y + 2, 2x2 – 3x +1
In the Standard Form of a Polynomial
The degrees of the monomials terms decrease from left to right.
Degree of a Polynomial
• Is the same as the degree of the monomial with the greatest exponent.
BinomialIs a polynomial of two terms.
Ex: 3c + 4
TrinomialIs a polynomial of three terms.
Ex: 2x4 + x2 - 2
Ex:
(5x2 + 3x +4) + (3x2 + 5)
8x2 + 3x +9
Adding Horizontally
Ex:
(7a2b3 + ab) +(1 – 2a2b3)
5a2b3 + ab +1
Adding Polynomials in
Columns
(2x4 – 5x2 + 4x + 5)+ (5x4 + 7x3 – 2x2 –2x)
2x4 + 0x3 – 5x2 + 4x + 5
5x4 + 7x3 – 2x2 – 2x + 0
2x4 + 0x3 – 5x2 + 4x + 5
5x4 + 7x3 – 2x2 – 2x + 0
7x4 + 7x3 – 7x2 + 2x + 5
Subtracting of Polynomials
Review : 2 polynomials are the additive inverses of each other if there sums equal zero.
4x7 – 7x - 5
Can be rewritten as
4x7 + (-7x) + (-5)
Key Questions•Is x-3 the additive inverse of x3?
No
01
0
33
33
xx
xx
•Is x-3 the additive inverse of –x3?
NO
01
0)(
33
33
xx
xx
•Is x-3 the additive inverse of –x-3?
•Yes
011
0)(
33
33
xx
xx
Ex 1: Find the Additive Inverse
7x4 - 3x + 5-7x4 + 3x – 5
Notice the signs changed
Subtract(5x2 + 3x – 2) – (2x2 + 1)
Change the sign of the terms in the 2nd
parenthesis.
Subtract(5x2 + 3x – 2) – (2x2 + 1)
5x2 + 3x – 2 – 2x2 – 1Subtract like terms.(remember you are
subtracting a 1)
Subtract(5x2 + 3x – 2) – (2x2 + 1)
5x2 + 3x – 2 – 2x2 – 1
Subtract like terms.
Subtract(5x2 + 3x – 2) – (2x2 + 1)
5x2 + 3x – 2 – 2x2 – 1
3x2 + 3x - 3
Subtract(2a2b2 + 3ab3 – 4b4) – (a2b2 – 5ab3 + 3b – 2b4)
Change the sign of every term in the 2nd parenthesis.
Subtract(2a2b2 + 3ab3 – 4b4) – (a2b2 – 5ab3 + 3b – 2b4)
2a2b2 + 3ab3 – 4b4 – a2b2 + 5ab3 – 3b + 2b4
Subtract(2a2b2 + 3ab3 – 4b4) – (a2b2 – 5ab3 + 3b – 2b4)
Subtract like terms.2a2b2 + 3ab3 – 4b4 – a2b2 + 5ab3 – 3b + 2b4
a2b2 + 8ab3 – 2b4 – 3b
Use Columns to Subtract
8x3 + 6x2 – 3x + 5 minus 5x3 – 3x2 – 2x + 4
8x3 + 6x2 – 3x + 5
Change the sign of each term in the 2nd part.
8x3 + 6x2 – 3x + 5
-5x3 + 3x2 + 2x – 4
3x3 + 9x2 – x + 1
Subtract using columns
2a4b + 5a3b2 – 4a2b3 minus 4a4b +2a3b2 – 4ab
2a4b + 5a3b2 – 4a2b3 + 0ab
-4a4b – 2a3b2 – 0a2b3 + 4ab
-2a4b + 3a3b2 – 4a2b3 + 4ab
9 – 2 Multiplying and Factoring
Multiplying a Monomial and a Trinomial
Ex 1: Simplify -2g3(3g3 + 6g – 5)Distribute -2g3 to each term inside the parenthesis.
( )(3g3) -2g2 ( ) –2g2 ( )
-2g2 (3g3) -2g2 (6g) –2g2(-5)
( )(3g3) -2g2 ( ) –2g2 ( )
-2g2 (3g3) -2g2 (6g) –2g2(-5)
Remember when multiplying you add exponents.
( )(3g3) -2g2 ( ) –2g2 ( )
-2g2 (3g3) -2g2 (6g) –2g2(-5)
Remember when multiplying you add exponents.
-6g2 + – 12g2 + +
( )(3g3) -2g2 ( ) –2g2 ( )
-2g2 (3g3) -2g2 (6g) –2g2(-5)
Remember when multiplying you add exponents.
-6g2 + 3 – 12g2+1 + 10g2
Finding the Greatest Common Factor
•GCF – what monomial is a factor in each term.
Ex 2: Find the GCF of 2x4 + 10x2 - 6x
• List the prime factors of each term.
2x4 = 2• x • • x •
10x2 = 2• • x •
6x = 2• x
2x4 = 2• x • x • x • x10x2 = 2• x • x • x6x = 2 • 3 • xWhich prime factors are in term?
2x4 = 2• x • x • x • x10x2 = 2• x • x • x6x = 2 • 3 • xWhich prime factors are in term? A ‘2’ and an ‘x’.
•The GCF is 2x.
•Do QC #2a – c, Pg. 501.
Factoring Out a Monomial
Find the GCF and then factor out the GCF from each term.
Factoring Out a Monomial Factor
• Ex: Factor 4x3 +12x2 – 16x
First find the GCF.
List the prime factors.
4x3 =
12x2 =
16x =
List the prime factors.
4x3 = 2• •x• • •
12x2 = 2 • 2 • • x • x
16x = 2 • • 2 • • x
List the prime factors.
4x3 = 2• 2 •x• x • x •
12x2 = 2 • 2 • 3 • x • x
16x = 2 • 2 • 2 • 2 • x
List the prime factors.
4x3 = 2• 2 •x• x • x •
12x2 = 2 • 2 • 3 • x • x
16x = 2 • 2 • 2 • 2 • x
The GCF is 4x.
Now Factor Out the GFC.
4x3 +12x2 – 16x
4x( ) + ( )(3x) + 4x(-4)
4x( x2 ) + ( 4x )(3x) + 4x(-4)
( ) (x2 + - )
(4x) (x2 + 3x - 4 )
Ex: 5x3 + 105 goes into each term
5(x3) + 5(2)
5(x3 + 2)
Ex: 6x3 + 12x2
6x2 goes into each term.
6x2 (x) + 6x2 (2)6x2 (x + 2)
Ex: 12u3v2 + 16uv4
4uv2 goes into each term.
4uv2(3u2) + 4uv2(4v2)
4uv2(3u2 + 4v2)
Ex: 18y4 – 6y3 + 12y2 6y2 goes into each term.
6y2(3y2) - 6y2(y) + 6y2(2)
6y2(3y2 – y + 2)
Ex: 8x4y3 + 6x2y4 2x2y3 goes into each term.
2x2y3(4x2) + 2x2y3(3y)
2x2y3(4x2 + 3y)
Ex: 5x3y4 + 7x2z3 + 3y2z
There is no common factor.
9 – 3 Multiplying Binomials
Using the Distributive Property
Ex 1: (4y + 5)(y + 3)
Using the Distributive Property
Ex 1: (4y + 5)(y + 3)
4y(y + 3) + 5(y + 3)
Using the Distributive Property
Ex 1: (4y + 5)(y + 3)
4y(y + 3) + 5(y + 3)
4y2 + 12y + 5y + 15
Using the Distributive Property
Ex 1: (4y + 5)(y + 3)
4y(y + 3) + 5(y + 3)
4y2 + 12y + 5y + 15
Combine like terms
Using the Distributive Property
Ex 1: (4y + 5)(y + 3)
4y(y + 3) + 5(y + 3)
4y2 + 12y + 5y + 15
Combine like terms
4y2 + 17y + 15
Use the Distributive Property
Ex 2: (3a – b)(a + 7)
3a(a + 7) – b(a + 7)
3a2 + 21a – ab – 7b
No like terms
Multiplying 2 Binomials
When multiplying 2 Binomials, a useful tool to use is the FOIL Method.
First terms
Outside terms
Inside terms
Last terms
•Ex: (x + 5)(x + 4)
(x + 5)(x + 4)
x2
First terms
(x + 5)(x + 4)
4x
Outside terms
(x + 5)(x + 4)
5x
Inside terms
(x + 5)(x + 4)
20
Last terms
x2 + 4x + 5x + 20
x2 + 9x + 20
Ex:(x + 2)(x + 3)
x2 + 3x + 2x + 6x2 + 5x + 6
Ex:
(3x + 2)(x + 5)3x2 + 15x + 2x + 10
3x2 + 17x + 10
Ex: (4ab + 3)(2ab2 +1)8a2b3 + 4ab + 6ab2 + 3
Will there always be like terms
after multiplying binomials?
NO
Using the Vertical Method
Ex 1: (7a + 5)(2a – 7)
9 – 4 Multiplying Special Cases
Multiplying a Sum and a Difference.
Product of (A + B) and (A - B)
5) Multiply (x – 3)(x + 3)
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
x2 – 9
x -3
x
+3
x2
+3x
-3x
-9
This is called the difference of squares.
x2
+3x-3x-9
Notice the middle terms
eliminate each other!
Multiply (x – 3)(x + 3) using (a – b)(a + b) = a2 – b2
You can only use this rule when the binomials are exactly the same except for
the sign.
(x – 3)(x + 3)
a = x and b = 3
(x)2 – (3)2
x2 – 9
Ex: Multiply: (y – 2)(y + 2)(y)2 – (2)2
y2 – 4
Ex: Multiply: (5a + 6b)(5a – 6b)
(5a)2 – (6b)2
25a2 – 36b2
Multiply (4m – 3n)(4m + 3n)
1. 16m2 – 9n2
2. 16m2 + 9n2
3. 16m2 – 24mn - 9n2
4. 16m2 + 24mn + 9n2
•The product of the sum and difference of two terms is the square of the first term minus the square of the second term.
(A + B)(A – B) A2 – B2
•When multiplying the sum and difference of two expressions, why does the product always have at most two terms?
The other terms are additive inverses of each other and cancel each other out.
EX: (r + 2)(r – 2)
r2 – 2r + 2r – 4
r2 – 4
Ex: (2x + 3)(2x – 3)
4x2 – 6x + 6x – 9
4x2 – 9
Ex: (ab + c)(ab – c)
a2b2 – c2
Ex:
(-3x + 4y)(-3x – 4y)
9x2 – 16y2
Squaring Binomials The square of a binomial is the square if the first term, plus or minus twice the product of the two terms, plus the square of the last term.
(A + B)2 = A2 + 2AB + B2
(A - B)2 = A2 - 2AB + B2
Common Mistake25)5( 22 xx
Common Mistake
2949
2547
252)52(
2Let x
25)5(
2
22
22
xx
Ex:1) Multiply (x + 4)(x + 4)
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
x2 +8x + 16
x +4
x
+4
x2
+4x
+4x
+16
Now let’s do it with the shortcut!
x2
+4x+4x+16
Notice you have two
of the same
answer?
1) Multiply: (x + 4)2
using (a + b)2 = a2 + 2ab + b2
a is the first term, b is the second term(x + 4)2
a = x and b = 4Plug into the formula
a2 + 2ab + b2
(x)2 + 2(x)(4) + (4)2
Simplify.x2 + 8x+ 16
This is the same answer!
That’s why the 2 is in the formula!
Ex: 2) Multiply: (3x + 2y)2
using (a + b)2 = a2 + 2ab + b2
(3x + 2y)2
a = 3x and b = 2y
Plug into the formulaa2 + 2ab + b2
(3x)2 + 2(3x)(2y) + (2y)2Simplify
9x2 + 12xy +4y2
Multiply (2a + 3)2
1. 4a2 – 9
2. 4a2 + 9
3. 4a2 + 36a + 9
4. 4a2 + 12a + 9
Ex: (x + 5) 2
(x2 + 2(x)(5) + 25)
x2 + 10x + 25
Ex Multiply: (x – 5)2
using (a – b)2 = a2 – 2ab + b2
Everything is the same except the signs!
(x)2 – 2(x)(5) + (5)2
x2 – 10x + 25
Ex Multiply: (4x – y)2
(4x)2 – 2(4x)(y) + (y)2
16x2 – 8xy + y2
Ex: (y – 3)2
y2 - 2(y)(3) + 9
y2 – 6y + 9
Ex: (2a – 3b)2 4a2 - 2(2a)(3b) + 9b2
4a2 - 12ab + 9b2
Multiply (x – y)2
1. x2 + 2xy + y2
2. x2 – 2xy + y2
3. x2 + y2
4. x2 – y2
9 – 5 Factoring trinomials of the type x2 + bx +c
To factor a trinomial of in the form of
x2 + bx + cWhere c > 0
We must find 2 numbers that:
A.) their product is c andB.) their sum or difference is b.
C. If the sign of c is (+), both numbers must have the same sign.
To figure out the 2 factors, we are to use the Try, Test, and Revise Method.
What two numbers have the sum of 5 and the product of 6?
We look at the factors of 6 that add up to 5.
Factors of 61 and 6
2 and 3
Factors of 61 and 6 sum is 7
2 and 3sum is 5
2,3
What two numbers have the sum of 8 and the product of 12?
We look at the factors of 12 that add up to 8.
1 and 12
2 and 6
3 and 4
1 and 12 sum is 13
2 and 6 sum is 8
3 and 4 sum is 7
2,6
What two numbers have the sum of -8 and the product of 7?
Since the sum is (-) and the product is (+).
Both numbers are (-).
-1 and -7
Only set of factors.
-1 and -7
sum is -8
Factor x2 + 8x + 12
b = 8, c = 12
What factors of 12 add up to 8?
Factors of 121 and 12 -1 and -12
2 and 6 -2 and -6
3 and 4 -3 and -4
2 and 6
(x + 2)(x + 6)
Factorx2 – 10x + 16 b = -10, c = 16
What factors of 16 add up to -10?
Factors of 161 and 16 -1 and 16
2 and 8 -2 and -8
4 and 4 -4 and -4
-2 and -8
(x – 2)(x – 8)
Factorp2 – 3pq + 2q2
b = -3, c = 2
What factors of 2 add up to -3?
1 and 2 -1 and -2
-1 and -2
(p – q)(p – 2q)
Factors of -2
To factor a trinomial of in the form of
x2 + bx + cWhere c < 0
Watch your signs most students have the right factors, but the wrong signs.
What 2 numbers have a sum of -6 and a product of -7?
-1 and 7 or 1 and -7
1 and -7
What 2 numbers have a sum of -2 and a product of -8?
-1 and 8 1 and -8-2 and 4 2 and -4
-2 and 4
What 2 numbers have a sum of 1 and a product of -2?
1 and -2 -1 and 2
What 2 numbers have a sum of 1 and a product of -2?
1 and -2 -1 and 2
Factor u2 – 3uv - 10v2
b = -3 and c = -10What factors of -10 have the sum of -3.
Factors of -101 and -10 -1 and 10
2 and -5 -2 and 5
2 and -5
(u + 2v)(u – 5v)
Factor x2 + 3x - 4
b = 3 and c = -4What factors of -4 have the sum of 3.
Factors of -41 and -4 -1 and 4
2 and -2
-1 and 4
(x – 1)(x + 4)
Factor y2 – 12yz – 28z2
b = -12 and c = -28What factors of -28 have the sum of -12.
Factors of -281 and -28 -1 and 28
2 and -14 -2 and 14
4 and -7 -4 and 7
2 and -14
(x + 2z)(x – 14z)
Review: (y + 2)(y + 4) Multiply using FOIL or using the Box Method.Box Method: y + 4
y y2 +4y + 2 +2y +8
Combine like terms.FOIL: y2 + 4y + 2y + 8
y2 + 6y + 8
1) Factor. y2 + 6y + 8Put the first and last terms into the
box as shown.
What are the factors of y2?
y and y
y2
+ 8
1) Factor. y2 + 6y + 8Place the factors outside the box as
shown.
y2
+ 8
y
y
What are the factors of + 8?
+1 and +8, -1 and -8
+2 and +4, -2 and -4
The second box works. Write the numbers on the outside of box for your solution.
1) Factor. y2 + 6y + 8Which box has a sum of + 6y?
y2
+ 8
y
y
y2
+ 8
y
y+ 1 + 2
+ 8 + 4
+ y
+ 8y + 4y
+ 2y
1) Factor. y2 + 6y + 8
(y + 2)(y + 4)Here are some hints to help you choose
your factors.
1) When the last term is positive, the factors will have the same sign as the middle
term.
2) When the last term is negative, the factors will have different signs.
x2
- 63
2) Factor. x2 - 2x - 63Put the first and last terms into the box
as shown.
What are the factors of x2?
x and x
2) Factor. x2 - 2x - 63 Place the factors outside the box as
shown.
x2
- 63
x
x
What are the factors of - 63?
Remember the signs will be different!
2) Factor. x2 - 2x - 63Use trial and error to find the correct
combination!
Do any of these combinations work?
The second one has the wrong sign!
x2
- 63
x
x
+ 21
- 3
+21x
-3x x2
- 63
x
x - 7
+ 9
-7x
+9x
2) Factor. x2 - 2x - 63Change the signs of the factors!
Write your solution.
(x + 7)(x - 9)
x2
- 63
x
x + 7
- 9
+7x
-9x
9 – 7 Factoring Special Cases
In 9 – 4 we used the square of binomials.
(A + B)2 = A2 + 2AB + B2
(A - B)2 = A2 - 2AB + B2
EX: (x + 8)2 or (x - 8)2
A = x and B = 8
A2 + 2AB + B2
(x)2 + 2(x)(8) + (8)2
x2 + 16x + 64
A = x and B = 8
A2 - 2AB + B2
(x)2 - 2(x)(8) + (8)2
x2 - 16x + 64
Perfect-Square Trinomials
A2 + 2AB + B2 =(A + B)(A + B) = (A + B)2
A2 - 2AB + B2 = (A - B) (A - B)
= (A - B)2
•The perfect-square trinomials are the reverse of the square binomials.
Ex 1: Where ‘A’ = 1
x2 + 10x + 25
A2 + 2AB + B2
A2 = x2 2AB = 10x B2 = 25
A = x and B = 5
(A + B)2
(x + 5)2
x2 + 10x + 25 = (x + 5)2
Always make sure the 1st and last terms are
perfect squares.
Ex 2: y2 – 22y + 121
Are y2 and 121 perfect squares?
Yes, both are perfect squares for y and 11
Since the middle term is negative, you subtract.
(y – 11)2
Ex 3: m2 + 6mn + 9n2
Perfect squares?
m2 m and 9n2 3n
(m + 3n)2
Ex 2: if A ≠ 1
64y2 + 48y + 9
Are 64y2 and 9 perfect squares?
YES
(8y)2 + 48y + (3)2
(8y + 3)2
The one thing you need to look at the middle term, is it
equal to 2AB.
2(8y)(3) = 48y
Ex: 16h2 + 40h + 25
(4h)2 + 40h + (5)2
Does 2AB = 40h?
2(4h)(5)
Yes, 40h
(4h + 5)2
Difference of Two Squares
A2 – B2 = (A + B)(A – B)
This is the reverse from section 9 – 4.
(A + B)(A – B) = A2 – B2
Make sure the A and B terms are perfect squares.
Ex: a2 – 16
Both are perfect squares a2 a and 16 4.
(a + 4)(a – 4)
Ex: m2 – 100 (m + 10)(m – 10)
Ex: 9b2 - 25Are both perfect squares?
Yes, 9b2 3b and 25 5
(3b + 5)(3b – 5)
Ex: 4w2 - 49(2w + 7)(2w – 7)
•Factor: 14a4 – 14a2
14a2(a2 – 1)
Difference of Squares
14a2(a + 1)(a - 1)
Factor:
3x4 + 30x3 + 75x2
3x2(x2 + 10x + 25)
Factor:
3x4 + 30x3 + 75x2
3x2(x2 + 10x + 25)
This is a binomial square.
3x2(x + 5)2
Factor:
2a4 + 14a3 + 24a2
2a2(a2 + 7a + 12)
2a2(a + 3)(a + 4)