chapter(6:( polynomials(and(polynomial( functions( 6... · in this lesson, you will design an open...

Chapter 6: Polynomials and Polynomial Functions

Chapter 6: Polynomials and Polynomial Functions Assignment Sheet

Date Topic Assignment Completed 2/1 6.1-‐ Using Properties of

Exponents pg. 326 # 17-‐27 (odd), 52, 54, 55, 57a, 57b

2/2 6.2-‐ Evaluating and Graphing Polynomial Functions

Begin Homework 6.2

2/3 6.2-‐ Evaluating and Graphing Polynomial Functions

Finish Homework 6.2

2/4 6.3-‐ Adding, Subtracting, and Multiplying Polynomials

Homework 6.3

2/5 6.4-‐ Factoring and Solving Polynomial Equations

pg. 349 # 34-‐84 (even)

2/8 6.4-‐ Factoring and Solving Polynomial Equations

pg. 349 # 87, 88, 90-‐95, 97-‐101

2/9 Review 6.1-‐6.4 pg. 388 # 1-‐4, 7-‐15 pg. 391 # 1-‐18, 28, 29, 32

2/10 Test 6.1-‐6.4 None 2/11 6.5 Day 1-‐ Long Division,

Synthetic Division, and the Remainder Theorem

2/16 6.5 Day 2-‐ Long Division and the Factor Theorem

pg. 356 # 19-‐23 (odd), 28-‐34 (even), 40-‐50 (even)

2/17 6.6-‐ Finding Rational Zeros pg. 363 # 34, 38, 40, 41, 43, 52, 56 2/18 6.7 Day 1-‐ The Fundamental

Theorem of Algebra pg. 369 # 22-‐46 (even)

2/19 6.7 Day 2-‐ The Fundamental Theorem of Algebra

pg. 369 # 21-‐41 (odd)

2/22 Review 6.5-‐6.7 pg. 389 # 16-‐19 pg. 391 # 19-‐28

2/23 Review 6.5-‐6.7 6.5-‐6.7 Review Packet 2/24 Test 6.5-‐6.7 None

6.1 -‐ Using Properties of Exponents Warm Up: Evaluate the following.

(−2)3 (−2)4 − 24 Rules for Exponents: Examples: 1.) Multiplying Exponents 32 ⋅33 = _________

2.) Exponents Raised to an Exponent (32 )3 = ____________

3.) Product Raised to an Exponent (ambn )k = amkbnk (4x2 )3 = ____________

4.) Quotient of Powers Property am

an= am−n

57

54 = ____________

5.) Zero Exponent Property a0 = 1, a ≠ 0 (5xy2 )0 = ____________

86

86 = ____________

6.) Negative Exponents 3−2 = ____________

75

77 = ____________

7.) Quotient Raised to a Power am

bn⎛⎝⎜

⎞⎠⎟

k

= amk

bnk

32

4⎛⎝⎜

⎞⎠⎟

4

= ____________

Evaluate the following exponential expressions: 1.) x8x3 = ______ 2.) (x8 )3 = ______ 3.) 96 ⋅92 = ____________ 4.) (96 )2 =__________ 5.) 4x3( )4=________ 6.) (23h19 )2 =__________

m n m na a a +⋅ =

( )nm mna a=

( ) 1mmaa

− =

7.) 2r5y3( )2( )3 = ____________ 8.) 3x2y3( )6 2x5y7( )4 =_______________

9.) 2x7

x4 =_______________ 10.) x

4

2x7 =____________ 11.) 3

9a5

311a10 =___________

12.) 8−2 =__________ 13.) x−6x2=__________ 14.) x−9y7=____________

15.) x6

x−2 =_________ 16.) x

−6

x2 =__________ 17.) −2x

−3y7z−8

x4y−1 =___________

18.) 2x3y−4( )−1 =_________ 19.) −2x−1yz−8x

⎛⎝⎜

⎞⎠⎟

2

=_________ 20.) (−12 ⋅4)−2

1−14−1 =____________

21.) 5x−2

y3⎛⎝⎜

⎞⎠⎟

−4

=________ 22.) 22 xy6( )−3 3x2y−3( )3 =___________

23.) −8x3y−1

(2x0y2 )3 =______ 24.) 5

−1x2y8

⋅ −2x−1y

x3y =_______________

25.) −2x2( )25

⋅15x−2

2x3 =_____________

6.2 -‐ Evaluating and Graphing Polynomial Functions

f (x) = 19x10 − 8x9 + 0.5x4 + x2 − 2 Characteristics of Polynomials:

1.) The degree is the highest exponent à (10) 2.) Leading coefficient is the coefficient for the term with the highest power à (19) 3.) All exponents are whole numbers à (0,1,2,3,..) 4.) All coefficients are REAL numbers

For the following:

1.) Is it a polynomial or not? If Yes- 2.) Put in standard form 3.) State the degree 4.) State the type: constant, linear, quadratic, cubic, quartic, quintic 5.) State the leading coefficient

f x = !

!x!-‐3x!-‐7 f x = x! + 3!

1.) 1.) 2.) 2.) 3.) 3.) 4.) 4.) 5.) 5.) f x = 6x! + 2x-‐! + x f x = -‐.05x+ πx!-‐ 2 1.) 1.) 2.) 2.) 3.) 3.) 4.) 4.) 5.) 5.)

___)(,___)(,

→−∞→→∞→xfxAsxfxAs

Polynomials are functions with varying degrees and graphical behavior 1.) f x = 4

x f(x) Remarks -‐1 0 y-‐intercept 1

Degree: Leading Coefficient: Type: 2) f x = 2x+ 4

x f(x) Remarks 0 y-‐intercept 0 x-‐intercept

Degree: Leading Coefficient: Type: End Behavior:

___)(,___)(,


As x→∞, f (x)→ ___As x→ −∞, f (x)→ ___

As x→∞, f (x)→ ___As x→ −∞, f (x)→ ___

3.) f x = -‐2x+ 1 X f(x) Remarks

Degree: Leading Coefficient: Type: End Behavior: 4.) f x = x!-‐8x+ 15

X f(x) Remarks Degree: Leading Coefficient: Type: End Behavior 5.) f x = -‐x! + 8x-‐15

X f(x) Remarks Degree: Leading Coefficient: Type:

End Behavior:

___)(,___)(,


___)(,___)(,


As x→∞, f (x)→ ___As x→ −∞, f (x)→ ___

6.) f x = x! + x!-‐4x-‐1

Degree: Leading Coefficient: Type: End Behavior: 7.) f (x) = −x3 − x2 + 4x +1

X f(x) Remarks

Degree: Leading Coefficient: Type: End Behavior: 8.) f(x) = x!-‐3x! + x

X f(x) Remarks Degree: Leading Coefficient: Type: End Behavior:

X f(x) Remarks

___)(,___)(,


___)(,___)(,


___)(,___)(,


9.) -‐x!-‐2x! + 2x! + 4x Degree: Leading Coefficient: Type: End Behavior 10.) f x = x!-‐3x!-‐x Degree: Leading Coefficient: Type: End Behavior 11.) f x = -‐3x! + 2x! + x Degree: Leading Coefficient: Type: End Behavior NOTE: Very tiny graph. Let the scale for the graph be 5 units = 1

End Behavior Even power, + lead coefficient

Even power, -‐ lead coefficient

Odd power, + lead coefficient

Odd power, -‐ lead coefficient

For the following: a.) Identify the degree (odd or even) b.) Describe left and right hand behavior (end behavior) 1. 𝑓 𝑥 = 𝑥 2. 𝑓 𝑥 = 𝑥! 3. 𝑓 𝑥 = −𝑥! − 2𝑥! − 𝑥 + 1 4. 𝑓 𝑥 = 2𝑥! + 2𝑥! − 5𝑥 + 1 5. 𝑓 𝑥 = 𝑥! − 3𝑥! + 2𝑥 − 1 6. 𝑓 𝑥 = −𝑥! + 𝑥! + 5𝑥 − 4

6.2 Homework 1. Decide whether the function is a polynomial. 2. If it is, write the function in standard form. 3. State the degree (1st, 2nd, 3rd etc.) , type (linear, quadratic, cubic, quartic etc.), and its leading coefficient.

a. f (x) = x − 5+ x3 2 b. f (x) = −6x2 + x − 3x

c. f (x) = 22 −19x + 2x d. f (x) = 36x2 − x3 + x4 Describe the end behavior of each polynomial function below WITHOUT the use of a graphing calculator. Use the rules of end behavior that we learned in class today. a. f (x) = −5x3 As x→ −∞, f (x)→ _____, x→∞, f (x)→ _____ b. f (x) = −x3 +1 As x→ −∞, f (x)→ _____, x→∞, f (x)→ _____ c. f (x) = 2x + 3x3 As x→ −∞, f (x)→ _____, x→∞, f (x)→ _____ d. f (x) = 2x2 − x As x→ −∞, f (x)→ _____, x→∞, f (x)→ _____ e. f (x) = −x4 + 3x3 As x→ −∞, f (x)→ _____, x→∞, f (x)→ _____ f. f (x) = x4 + 2 As x→ −∞, f (x)→ _____, x→∞, f (x)→ _____ g. f (x) = 4x + 2 As x→ −∞, f (x)→ _____, x→∞, f (x)→ _____ h. f (x) = 6 − x As x→ −∞, f (x)→ _____, x→∞, f (x)→ _____

Graph each of the following functions using your graphing calculator. Fill in the required information in each table. Then state the degree, leading coefficient and type of function, and end behavior for each function

1. ) f (x) = − 12x2 + x + 4

2.) f (x) = −x3 + 3x

3.) f (x) = x3 − 3x −1

x f(x) Critical Points x-intercept x-intercept y-intercept vertex

x f(x) Critical Points x-intercept x-intercept x-intercept y-intercept local maximum local minimum

x f(x) Critical Points x-intercept x-intercept x-intercept y-intercept local maximum local minimum

4.) f (x) = x4 − 4x3 + 2x

5.) f (x) = −x4 − 4x3 + 2x

6.) f (x) = x5 + 3x3 − x

x f(x) Critical Points x-intercept x-intercept x-intercept x-intercept y-intercept local maximum local minimum local minimum

x f(x) Critical Points x-intercept x-intercept x-intercept x-intercept y-intercept local maximum local maximum local minimum

x f(x) Critical Points x-intercept x-intercept x-intercept y-intercept local minimum local maximum

6.3 -‐ Adding, Subtracting, and Multiplying Polynomials Warm Up: Multiply, add, or subtract the following polynomials. 1.) )5(2 −x 2.) )15(4 2 +−− xx 3.) )2)(3( xx 4.) 22 127 mm − Add, Subtract or Multiply the following (consider using a vertical or a horizontal method): 1.) =+−+−−+ )810()723( 2323 xxxxx 2.) =−++ )135)(2( 2 xxx 3.) =−+−+ )43()32( 22 xxxx 4.) =−++− )5)(12( 233 xxxx

5.) )1)(3)(12( +++ xxx 6.) )1)(52)(32( −−− xxx 7.) =+ 2)6( yx 8.) ( ) =− 353x

In this lesson, you will design an open box out of grid paper that is 20 grids by 18 grids. The box will be formed by making the square cuts shown in the diagram and folding up the sides. x x x x x x x x x Volume of a Box = (Length)(Width)(Height)

x 1 2 3 4 5 6 7 8 9 Volume 1.) Which value of x maximizes the volume of the box? 2.) Are there other possible values of x that that could provide a larger volume? 3.) Provide the formula for the volume of the box in terms of x: Volume of the box = 4.) Graph the function above on your graphing calculator. Sketch the part of graph below that makes sense in the context of the problem. 500 What is the domain for this problem? What is the range for this problem? x 5.) Use your calculator to find the value of x that maximizes the volume of the box.

VOLUME

6.3 Homework Multiply, Add or Subtract the following polynomials. 1.) −(x3 − 3x2 + 5x −1)+ (2x3 − 5x + 8) = 2.) (x − 3)(−2x2 − x + 5) = 3.) (−3x3 − 2x2 + x)− (5x2 + 4x − 9) = 4.) (2x3 − x + 3)(−x3 + x2 −1) = 5.) (x + 4)(2x + 5)(3x + 4) 6.) (x − 3)(2x +1)(2x − 4) 7.) (x − 3y)2 = 8.) x −1( )3 =

You are designing a box to be made of a piece of cardboard that is 18 inches by 16 inches. The box will be formed by making the square cuts shown in the diagram and folding up the sides. x x x x x x x x x Volume of a Box = (Length)(Width)(Height) Find the volumes of the boxes that can be created for the given values of x in the table

x 1 2 3 4 5 6 7 8 Volume 1.) Which integer value of x maximizes the volume of the box? 2.) Are there other possible values of x that that could provide a larger volume? 3.) Provide the formula for the volume of the box in terms of x. VOLUME =

4.) Write the formula from #3 in the form

�

y = ax 3 + bx 2 + cx + d 5.) Graph the function above on your graphing calculator. Sketch the part of graph below that makes sense in the context of the problem. 500 What is the domain for this problem? What is the range for this problem? x 6.) Use your calculator to find the value of x that maximizes the volume of the box.

VOLUME

1.) 125 + x3 5.) bx2 + 2a + 2b + ax2

2.) 64a4 − 27a 6.) 25x4 − 36

3.) x2y2 − 3x2 − 4y2 +12 7.) 4x6 − 20x4 + 24x2

4.) x3 − 2x2 − 9x +18 8.) a2b2 − 8ab3 +16b4

6.4 -‐ Factoring and Solving Polynomial Equations Warm up: FACTOR

RECALL:

a2 − b2 = (a − b)(a + b)a3 − b3 = (a − b)(a2 + ab + b2 )a3 + b3 = (a + b)(a2 − ab + b2 )

35 14242 xxx =+

0182 5 =− yy

281273 23 =−+ xxx

Solve the equations by : 1. Set the equation equal to zero. 2. Factor. 3. Set the factors equal to zero and solve for the variable. OR 1. Isolate the variable raised to the nth power. 2. Take the nth root of both sides of the equation. Example 1: Find all real number solutions for x Example 2: Find all real number solutions for y Example 3: Find all real number solutions for x

254 3 −=x

04129 24 =+− xx

Example 4: Find all real number solutions for x Example 5: Find all real number solutions for x

6.5 Day 1: Long Division, Synthetic Division and the Remainder Theorem Warm Up:

115620 For the problem below find the missing factor. Check your answer by using long division to make sure it is correct. 1.) )(_______)9(6322 −=−− xxx 2. x2 + 2x − 35 = (x − 5)(_______)

3.) x2 +11x +10 = (x +1)(_______) 4.) x2 + x − 56 = (x − 7)(_______)

Use LONG DIVISION of polynomials to divide the following: 1.) Divide y4 + 2y2 − y + 5 by y2 − y +1( ) 2.) Divide 1532 34 −++ xxx by x2 − 2x + 2( ) 3.) )3()9145( 234 xxxxx +÷++

Synthetic Division (Also called Synthetic Substitution) Synthetic division is an alternate to polynomial long division when the divisor is first degree. 1) 82)( 23 +−−= xxxxf divide f(x) by (x + 2) 2) f (x) = x3 − 7x − 6 divide f(x) by (x -‐ 2) 3) f (x) = 10x4 + 5x3 + 4x2 − 9 divide f(x) by (x + 1)

Remainder Theorem: Let 82)( 23 +−−= xxxxf

a.) Use long division to divide f (x) by x + 2( ) . What is the quotient? What is your remainder?

b.) Evaluate )2(−f for the above function. How does your answer relate

to the above question? Remainder Theorem If a polynomial f (x) is divided by (x − k) then the remainder is f (k) . Use the remainder theorem to find the remainder of the division problem.

5223)( 23 −+−= xxxxf by x − 2 . The remainder will be f (2) .

Divide: )3()962()( 23 +−−+= xbyxxxxf . The remainder will be f (−3)

6.5 Day 2: Long Division and the Factor Theorem Warm Up: Use long division and synthetic division to divide 82)( 23 +−−= xxxxf by )1( −x .

Use synthetic division to divide the following polynomials: 1.) Find )2()67( 3 −÷−− xxx Is x-‐2 a factor of ?673 −− xx Why or why not? ______________ =)2(f _______________ 2.) Find )1()94510( 234 +÷−++ xxxx Is x+1 a factor of ?94510 234 −++ xxx Why or why not? ____________ f(-‐1) =_____

3.) Determine )4()416( 24 +÷++− xxxx Is x+4 a factor of ?416 24 ++− xxx Why or why not? _________________ f(-‐4)= Factor Theorem A polynomial )(xf has a factor (x − k) IF and ONLY IF f (k) = 0 . Fully factor the functions below given one zero/x-‐intercept of the function, then find the other zeros. 1.) 82133)( 23 −++= xxxxf given that 0)4( =−f 2.) 4056142)( 23 −−−= xxxxf given that 0)10( =f

3.) 184714)( 23 −+−= xxxxf given that 9=x is a zero of the function. 4.) 155294)( 23 +−+= xxxxf given that 5−=x is a zero of the function.

6.6 -‐ Finding Rational Zeros The POSSIBLE rational zeros of the above function can be summarized as the factors of the constant in the function, often denoted by the letter p, divided by the factors of the lead coefficient, often denoted by the letter q. Example 1 Given the following polynomial function: 5252)( 23 +−−= xxxxf Factors of p = 5 : Factors of q = 2 : factors of pfactors of q

:

Use synthetic division to find any rational zeros of the function 5252)( 23 +−−= xxxxf .

Example 2 Given the following polynomial function: 30114)( 23 +−−= xxxxf Factors of p = 30 : Factors of q = 1 : factors of pfactors of q

:

Use synthetic division to find any rational zeros of the function. Example 3 Given the following polynomial function, find the rational zeros:

211113)( 234 −+++= xxxxxf

Example 4 Given the following polynomial function: 65176)( 23 +−−= xxxxf Find the rational zeros. NOTE: The Rational Zero Theorem only identifies rational zeros! If the zeros of the polynomial are irrational or imaginary, then the Rational Zero Theorem is of no help.

6.7 -‐ The Fundamental Theorem of Algebra (FTOA) When all real and imaginary solutions are counted, a polynomial of degree n has exactly n solutions when you count both the real and imaginary solutions. Quadratic Equations: Degree =_____________ Find ALL zeroes of the equations below: 1.) 42 −= xy 2.) 42 += xy Solutions:________ Solutions:___________ 3.) 122 +−= xxy Solutions: _____________ Quadratic Equations will always have a total of ________ real and imaginary solutions.

Cubic Equations: Degree = _______________ Find all zeroes of the equations below: 1.) xxxy 223 −−= Solutions: ______________ 2.) 48163 23 +++= xxxy Solutions:______________ 3.) 1523 +−+= xxxy Solutions: ____________ Cubic Equations will always have a total of ________ real and imaginary solutions.

Quartic Equations Degree=___________ Find all zeroes of the equations below: 1.) 6555 234 −−++= xxxxy Solutions:____________ 2.) 842 234 −−+−= xxxxy Solutions:_____________ 3.) 1224 −+= xxy Solutions:_______________ Quartic Equations will always have a total of ________ real and imaginary solutions.

Quintic Equations Degree:___________

61382 245 +−+−= xxxxy Quintic Equations will always have a total of ________ real and imaginary solutions.

Writing Polynomial Equations Write the polynomial function of the least degree that has real coefficients, the given zeroes and a leading coefficient of 1.

f (x) = (x − r1)(x − r2 )(x − r3)...

r1,r2,r3... are the zeros of the function

1.) -‐6, 3, 5 f (x) = (x − (−6))(x − 3)(x − 5)

2.) 2,-‐2,-‐6i f (x) = (x − 2)(x − (−2))(x − (+6i)(x − (−6i))

3.) 5, 2+3i

4.) 4, 7, 3-2i

chapter(6:( polynomials(and(polynomial( functions( 6... · in this lesson, you will design an open...

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