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PRE-CALCULUS 12 Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES

This handout is review supplement for Chapter 3 Polynomial Functions in the Pre-calculus 12 textbook. Complete each of these review exercises. Refer to the notes as needed. Check your results with those in the answer key.

Mr. Yuill, 2017

3.1 Characteristics of Polynomial Functions

Points to Consider

• Polynomials are made up of terms that are added or subtracted.

• The coefficient of each term must be a real number.

• The exponent of any variable must be a non-negative integer.

• The degree of a polynomial is the greatest degree of all terms in that polynomial.

• The leading coefficient is the coefficient of the term with the greatest degree.

• You should be able to identify which type of polynomial function produces a graph,

based on things like maximum possible number of x-intercepts and end behaviours.

1. Determine whether or not each of the following functions is a polynomial function. If

not then briefly show or explain why not.

a) 2 53 4 1y x x

x= − + − b) 4 3 2( ) 3 8 4 9f x x x x x= − + − + −

c) 5 4 323

( ) 2 0.75 7 5 2g x x x x x= + − + − d) 3 214

5 4 1y x x x x x=− + − − +

2. What are the degree, type, leading coefficient, and constant term of each polynomial

function?

a) 3 27 11 2 12y x x x= − − +

b) 5 4 3 25 12 2

( ) 2 4 8f x x x x x x=− + − + − −

3. For each polynomial function describe the following things about its graph:

i) the end behavior, ii) the maximum number of x-intercepts, and iii) the y-intercept.

a) 4 3 2( ) 3 5 8 14g x x x x x= − + − − +

b) 5 4 3 22 8 9 6 1y x x x x x= − − + + −

2 Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES

4. Determine the following things about the graph below.

a) Whether the graph represents an odd- or even-degree polynomial function.

b) Whether the leading coefficient of the polynomial is positive or negative.

c) The number of x-intercepts.

d) The domain and range of the polynomial function.

5. Determine the following things about the graph below.

a) Whether the graph represents an odd- or even-degree polynomial function.

b) Whether the leading coefficient of the polynomial is positive or negative.

c) The number of x-intercepts.

d) The domain and range of the polynomial function.

Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES 3

6. The cost in dollars of manufacturing n items is given by the function 2( ) 1500000 100C n n n= + + .

a) Determine what kind of polynomial function it is.

b) Draw a rough sketch of what the graph should look like.

c) Determine the cost of manufacturing 500 items.

3.2 The Remainder Theorem

Points to Consider

• When you divide a polynomial P(x) by binomial x – a, where a is a constant integer, the

result of ( ) ( )P x x a − or ( )P x

x a− equals ( )

RQ x

x a+

−, where Q(x) is the quotient, and R is

the remainder.

• The equation ( ) ( ) ( )R

P x x a Q xx a

− = +−

can be rewritten as ( ) ( ) ( )P x x a Q x R= − + .

• Before division takes place the terms of ( )P x must be written in descending order of

degree (from greatest-degree term to least-degree term).

• Before division takes place any “missing” terms must be replaced by 0 nx+ .

• The leading coefficient is the coefficient of the term with the greatest degree.

• The Remainder Theorem states that when ( )P x is divided by x – a then the remainder R

equals ( )P a .

1. Divide. Write the result in the ( )R

Q xx a

+−

form. Identify any restrictions in the variable.

4 3 2( 3 7 4 1) ( 5)x x x x x− + + − −

2. Divide. Write the result in the ( )R

Q xx a

+−

form. Identify any restrictions in the variable.

2 3 42 3 7 2

2

x x x x

x

− + + − +

+

4 Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES

3. Divide. Write the result in the ( )R

Q xx a

+−

form. Identify any restrictions in the variable.

5 4 2(5 12 90 140 17) ( 4)n n n n n+ + − + +

4. Use the Remainder Theorem to determine the remainder when 4 3 22 8 5x x x x− + − + is

divided by 2x− .

5. Use the Remainder Theorem to determine the remainder when 4 3 22 5 6 11x x x x− − + +

is divided by 3x+ .

6. Use the Remainder Theorem to determine the remainder when 6 4 3 23 2 4 6x x x x− + − −

is divided by 1x+ .

7. Suppose that 3 22 3 7x x kx− + − is divided by 1x+ . Determine what the value of

coefficient k must equal if the remainder resulting from the division equals 7− .

8. Suppose that 5 4 33 5 4 12x x kx x− − + + is divided by 2x− . Determine what the value of

coefficient k must equal if the remainder resulting from the division equals 4.

9. Suppose that 22 5 1x x− − is divided by x k− , where k is a constant integer. Determine

what the value of k must equal if the remainder resulting from the division equals 11.

10. The volume of a right rectangular prism is represented by the polynomial 3 26 9 14x x x+ − − . (Recall that volume V l w h= , where V is volume, l is length, w is

width, and h is height.) Determine if the binomial 4x+ could represent one of the

dimensions of the prism. How can you tell?

Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES 5

3.3 The Factor Theorem

Points to Consider

• If polynomial P(x) is divided by a binomial x a− , and the remainder R equals 0, then

x a− is a factor of P(x).

• Recall that the Remainder Theorem states that R equals ( )P a .

• The Factor Theorem states that x a− is a factor of a polynomial ( )P x if and only if

( ) 0P a = , which implies that 0R = .

• The Integral Zero Theorem states that if x a− is a factor of polynomial ( )P x (which has

coefficients that are integers) then a is an integer that is a factor of the constant term of

( )P x .

1. Determine whether or not 3x− is a factor of polynomial 3 210 29 30x x x− + − .

2. Determine whether or not 6x+ is a factor of polynomial 4 3 24 13 2 24x x x x+ − − + .

3. Suppose that (8) 0P = . Write the corresponding binomial factor of polynomial ( )P x .

4. Suppose that ( 4) 0P − = . Write the corresponding binomial factor of polynomial ( )P x .

5. Completely factor the polynomial 3 28 5 14x x x− + − − .

6. Completely factor the polynomial 4 3 22 3 19 6 8x x x x− − − + .

7. Completely factor the polynomial 3 64x − .

8. Completely factor the polynomial 5 4 3 27 7 23 44 20x x x x x+ + − − − .

9. Show that the polynomial 3 125x− − will equal 2( 5)( 5 25)x x x− + − + .

10. Determine what the zeros of the polynomial 3 2 8 12x x x+ − − are.

11. Determine what the zeros of the polynomial 4 3 23 7 15 18x x x x− − + + are.

12. Determine what the value of the constant term k must equal if x k− is a factor of

polynomial 24 5 21x x+ − .

6 Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES

13. Determine what the value of coefficient k must equal if 5x− is a factor of polynomial 3 26 30x x kx− + + .

14. Suppose that the volume of a right rectangular prism is represented by the polynomial 3 22 11 12x x x+ − − . One dimension of the prism equals 3x− . Determine what the other

two dimensions (binomials) of the prism must be.

15. Suppose that the volume of a right rectangular prism is represented by the polynomial 3 7 6x x− − . Determine what the dimensions of the prism are, written as binomials.

16. The product of four integers is represented by the polynomial 4 3 24 15 58 40x x x x+ − − − . Completely factor the polynomial, then determine what

those four integers are when 10x = .

Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES 7

3.4 Equations and Graphs of Polynomial Functions

Points to Consider

• A graph of a factorable polynomial can be sketched by plotting the zeros of the

polynomial as x-intercepts as well as determining the y-intercept by letting x = 0.

• Solving an equation based on a polynomial function can be solved by graphing both

sides of the equation and determining the x-coordinate of each point of intersection.

Alternately, the equation can be solved by zeroing one side of the equation and

determining the zeros of the polynomial on the other side.

• The multiplicity of a zero equals the exponent of the binomial ( )x a− that the zero is

associated with.

1. Analyze the polynomial-function graph below and determine the following things.

a) The sign of the leading term.

b) The zeros of the polynomial function, and the multiplicity of each.

c) The polynomial ( )P x , in factored and expanded form, that the graph represents.

(The leading coefficient will either equal 1 or –1.)

d) The roots of equation ( ) 0P x = .

e) The interval(s) where the function is positive and the interval(s) where the function

is negative.

8 Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES

2. Sketch the graph of 2( 4)( 2)y x x x= − + − the grid below.

Then state what the roots of the equation 2( 4)( 2) 0x x x− + − = are.

3. Sketch the graph of 3 28 9 18y x x x= − + + .

Then determine the roots of 3 28 9 18 0x x x− + + =

Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES 9

4. Solve the equation 4 3 23 23 33 14 0x x x x− − − − = , where x is a real number.

5. Solve the equation 4 3 25 3 2 9x x x x x+ + + + = + , where x is a real number.

6. A rectangular piece of land has dimensions 14 m by 11 m. A sidewalk with constant

width was built on all four sides of the land. The land remaining in the center, which is

now surrounded by the sidewalk, was dug up to create a pond with an area of 40 m².

Create an equation made up of a polynomial function, then solve the equation to

determine how wide the sidewalk must be. Reject any solution that is impossible.

7. A right rectangular prism is supposed to have a length of 8 m and width and height of

3 m. Changes to the dimensions are proposed, so that the width and height will be

increased by a certain number of metres, and the length will be decreased by the same

number of metres. This will cause the volume of the prism to be increased by 78 m³. By

how many metres will each dimension change? (By how many metres will the width and

height be increased and the length increased?) Reject solutions that make no sense

relative to the original dimensions.

8. The product of three consecutive integers equals –120. Determine what those three

integers are by creating and solving a polynomial equation that represents this situation.

ALSO…

• Try to complete some or all of the Chapter 3 Review exercises #1 to #15 on pages 153

and 154.

• Try to complete some or all of the Chapter 3 Practice Test exercises #1 to #10 on pages

155 and 156.

10 Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES

Answer Key

3.1 Characteristics of Polynomial Functions

1. a) NOT a polynomial. 155x

x−− = − , a term with a variable with a negative exponent.

b) & c) IS a polynomial

d) NOT a polynomial. 3

25 5x x x− =− and 1

24 4x x− =− , terms with variables

with fractional exponents.

2. a) Degree = 3, Type: cubic, Leading coefficient = 7, Constant term = 12

b) Degree = 5, Type: quintic, Leading coefficient = –1, Constant term = 12

8−

3. a) i) The graph extends down into quadrant 3 on the left, and down into

quadrant 4 on the right.

ii) 4 iii) 14

b) i) The graph extends down into quadrant 3 on the left, and up into

quadrant 1 on the right.

ii) 5 iii) –1

4. a) Even-degree polynomial b) Positive first term c) 4 x-intercepts

d) Domain = { | }x x R , Range = { | 134.53, }y y y R −

5. a) Odd-degree polynomial b) Negative first term c) 5 x-intercepts

d) Domain = { | }x x R , Range = { | }y y R

6. a) Quadratic function b)

c) $1 800 000

3.2 The Remainder Theorem

1. 3 2 4442 17 89

5x x x

x+ + + +

−, Restriction: 5x

2. 3 2 443 4 7 21

2x x x

x− + − +

+, Restriction: 2x−

3. 4 3 2 315 8 32 38 12

4n n n n

n− + − + −

+, Restriction: 4n−

4. 7−

5. 137

6. 10−

Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES 11

7. 5

8. 4

9. 4 (Reject 32

.)

10. 4x+ does NOT represent a dimension of the prism. ( 4) 54 0P − = , so 4x+ cannot

be a factor of the polynomial that represents the volume.

3.3 The Factor Theorem

1. 3x− is NOT a factor. ( 6 0R=− )

2. 6x+ IS a factor. ( 0R = )

3. 8x−

4. 4x+

5. 3 28 5 14 ( 1)( 2)( 7)x x x x x x− + − − = − + − −

6. 4 3 22 3 19 6 8 ( 1)( 2)(2 1)( 4)x x x x x x x x− − − + = + + − −

7. 3 264 ( 4)( 4 16)x x x x− = − + +

8. 5 4 3 2 27 7 23 44 20 ( 1) ( 2)( 2)( 5)x x x x x x x x x+ + − − − = + − + +

9. How this is done may vary from one student to another. But you should start by

rewriting the polynomial as 3 2( 0 0 125)x x x− + + + and then applying the Integral

Zero Theorem and Factor Theorem.

10. –2 and 3

11. –2, –1, and 3

12. 34

1 or 3k = −

13. 1k =−

14. The other dimensions will be ( 1)x + by ( 4)x + .

15. First rewrite the polynomial as 3 20 7 6x x x+ − − .

The dimensions are ( 2)x + by ( 1)x + by ( 3)x − .

16. 4 3 24 15 58 40 ( 5)( 2)( 1)( 4)x x x x x x x x+ − − − = + + + −

When 10x = , the integers must be (in ascending order) 6, 11, 12, and 15.

12 Chapter 3—Polynomial Functions REVIEW EXERCISES AND NOTES

3.4 Equations and Graphs of Polynomial Functions

1. a) Negative b) –4 (Mult. = 1), –2 (Mult. = 1), 1 (Mult. = 2)

c) 2 4 3 2( ) ( 4)( 2)( 1) 4 9 10 8P x x x x x x x x= − + + − = − − + + −

d) The roots of 2( 4)( 2)( 1) 0x x x− + + − = are –4, –2, and 1.

e) The function is positive for interval 4 2x− − .

The function is negative for intervals 4x − , 2 1x− , and 1x .

2. The sketch of the graph of 2( 4)( 2)y x x x= − + − should look like the graph below.

The roots of the equation 2( 4)( 2) 0x x x− + − = are –4, 0, and 2.

3. The sketch of the graph of 3 28 9 18y x x x= − + + should look like the graph below.

The roots of the equation 3 28 9 18 0x x x− + + = are –1, 3, and 6.

4. 2, 1, or 7x = − −

5. 1, or 1x =− are the only real roots of the equation.

6. The sidewalk width must be 3 m.

7. The dimensions will be changed by either 2 m or 39 m. The width and height will

be increased by either 2 m or 39 m, and the length will be decreased by 2 m or

39 m.

8. The three consecutive integers are –6, –5, and –4.