Foundations of Algebra 2 Name:

Unit 5 – Day 1 – Evaluating and Terminology of Polynomial Functions Period: Date:

Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading

coefficient.

Polynomial # of terms Standard Form? Degree Leading

Coefficient

2( ) 8f x x

4( ) 6 8 3f x x x

4( ) 6g x x

35( ) 10 3

2h x x x

3 2( ) 10 5 1h x x x

Use direct substitution to evaluate the polynomial function for the given value of x.

6. 4 2 3( ) 8 5 3 ; 2f x x x x x x 7.

3 5( ) 4 2 ; 3g x x x x

8. 3( ) 6 25 20; 5h x x x x 9.

4 31 3( ) 10; 4

2 4g x x x x x

Use synthetic substitution to evaluate the polynomial function for the given value of x.

10. 4 3 2( ) 8 12 6 5 9; 2f x x x x x x 11.

3 2( ) 8 7 35; 6g x x x x x

12. 3( ) 8 14 35; 4h x x x x 13.

4 3( ) 2 3 8 13; 2f x x x x x

14. 5 3( ) 6 10 27; 3g x x x x 15.

3 2( ) 7 11 4 ; 3h x x x x x

16. 4( ) 3 20; 4f x x x x

Describe and correct the error n evaluating the polynomial function 4 2( ) 4 9 21 7f x x x x when 2x .

17.

Foundations of Algebra 2 Name:

Unit 5 – Day 2 – Add, Subtract and Multiply Polynomials Period: Date:

Find the sum or difference

1. 2 2( 3 5) ( 4 8 9)x x x x 2. 2 2( 5 7) (5 11 8)z z z z

3. 2 3 2(2 8) ( 4 12 4)a a a a 4. 3 2 2(4 11 4 ) ( 7 5 8)t t t t t

5. 2 4 4 3(3 6 5 6 ) (5 6 4 )y y y y y y 6. 4 2 3 2(8 2 4) (3 12 8 )v v v v v v

7. 2 4 4 3(7 8 11 12 ) (8 11 13 )t t t y y y 8. 4 5 3 2(9 2 11 7) (8 2 8 )m m m m m m

9.

10.

Let 𝑓(𝑥) = 3(𝑥 − 5)2 and 𝑔(𝑥) = −2(𝑥 + 6)2. Which of the following is equivalent to

𝑓(𝑥) + 𝑔(𝑥)?

A. 4𝑥2 + 4𝑥 + 1 C. 𝑥2 − 147

B. 𝑥2 − 54𝑥 + 3 D. 2𝑥 − 54

11.

Which expression must be subtracted from (4𝑥3 − 5𝑥2 + 6𝑥 − 3) to result in

(6𝑥3 − 2𝑥2 + 4𝑥 + 7) ?

A. −2𝑥3 − 3𝑥2 + 2𝑥 − 10 C. −2𝑥3 − 7𝑥2 + 10𝑥 + 4

B. 2𝑥3 + 3𝑥2 − 2𝑥 + 10 D. 2𝑥3 − 7𝑥2 + 10𝑥 + 4

Foundations of Algebra 2 Name:

Unit 5 – Day 3 – Multiply Polynomials using Special Product Patterns Period: Date:

Describe and correct the error in simplifying the expression.

1.

Find the product of the expressions.

2. ( 5)( 5)x x 3. 2( 3 5)( 5)x x x 4.

2( 9)w

5. 2 2( 3 5)( 3 1)x x x x 6.

2( 4)y 7. 4 2 3( 7 )( 5)y y y y

8. 2(2 5)c 9.

2(3 4)t 10. (5 3)(5 3)p p

11. 3(7 )x y 12. (2 9 )(2 9 )a b a b 13.

3(3 7 )x y

14. ( 6)( 6) 8x x x

15. Find the product: (𝑥 − 1)(𝑥 + 5)(𝑥 − 3)

A. 𝑥3 + 15 C. 𝑥3 − 3𝑥2 − 13𝑥 − 15

B. 𝑥3 + 𝑥2 − 17𝑥 + 15 D. 𝑥3 + 7𝑥2 − 7𝑥 + 15

Foundations of Algebra 2 Name:

Unit 5 – Day 4 – Factoring Perfect Squares and Cubes Period: Date:

Factor the polynomials completely.

1. 2 1x 2. 2 100x 3. 24 100x

4. 27 28x 5. 22 144x 6. 5 33 48y y

7. 3 64y 8. 3 216x 9. 33 3x

10. 35 320z 11. 532 2z z

Factor the polynomials completely.

12. 4 25x

13. 4 23 12x x

14. 7 42 16x x

15. 5 410w w

Foundations of Algebra 2 Name:

Unit 5 – Day 5 – Polynomial Synthetic Division Period: Date:

Use Polynomial Synthetic Division to divide the following

1. 2(2 7 10) ( 5)x x x 2.

2(4 13 5) ( 2)x x x

3. 2( 8 1) ( 4)x x x 4.

2( 9) ( 3)x x

5. 3 2( 5 2) ( 4)x x x 6.

3( 4 6) ( 3)x x x

7. 4 3 2( 5 8 13 12) ( 6)x x x x x 8.

4 3( 4 16 35) ( 5)x x x x

Describe and correct the error in these problems

9.

Mixed Review: Factor and solve using X-Box / Shortcut

Solve the equations

11. 2 3 40 0x x 12.

25 13 6 0x x

13. 2 7 2 0x x 14.

24 15 10 0x x

Foundations of Algebra 2 Name:

Unit 5 – Day 6 – Factoring given a factor Period: Date:

Given polynomial f(x) and a factor of f(x), factor f(x) completely.

1. 3 2( ) 10 19 30; 6f x x x x x 2.

3 2( ) 2 40 64; 8f x x x x x

3. 3 2( ) 18 95 150; 10f x x x x x 4.

3 2( ) 9 8 60; 2f x x x x x

5. 3 2( ) 3 2 61 20; 5f x x x x x 6.

3 2( ) 2 15 34 21; 1f x x x x x

7. 3 2( ) 4 25 154 40; 10f x x x x x 8.

3 2( ) 3 4 28 16; 2f x x x x x

Foundations of Algebra 2 Name:

Unit 5 – Day 7 – Solving Given a Factor Period: Date:

If given polynomial f(x) and a factor of f(x), factor f(x) completely. If given polynomial function f(x) and a zero of f(x) find the

other zeros.

1. 3 2( ) 2 21 18f x x x x ; -3

2. 3 2( ) 8 42f x x x x ; ( 7)x

3. 3 2( ) 10 81 71 42f x x x x ;7

4. 3 2( ) 3 28 69 20f x x x x ; 5 5.

3 2( ) 2 5 196 99f x x x x 1

;2

6. 3 2( ) 11 34 104f x x x x ; 13 7.

3 2( ) 14 103 111 18f x x x x ; 6

8. 3 2( ) 2 10 71 9f x x x x ;9 9.

3 2( ) 2 5 32 80f x x x x ; 2 5x

Foundations of Algebra 2 Name:

Unit 5 – Day 8 Listing Possible Zeros for Polynomial Equations Period: Date:

Find all of the possible roots of the following polynomials.

1. 4 3 2( ) 6 7 6 8f x x x x x 2.

3 2( ) 5 4 20f x x x x

Factors of the constant term: Factors of the constant term:

Factors of the leading coefficient: Factors of the leading coefficient:

Possible rational zeros: Possible rational zeros:

Simplified list of zeros: Simplified list of zeros:

3. 4 2( ) 9 4 12f x x x x 4.

3 2( ) 7 15 9f x x x x

Factors of the constant term: Factors of the constant term:

Factors of the leading coefficient: Factors of the leading coefficient:

Possible rational zeros: Possible rational zeros:

Simplified list of zeros: Simplified list of zeros:

5. 3 2( ) 2 2 3f x x x x 6.

4 3 2( ) 4 4 64 192f x x x x x

7. 3 2( ) 2 7 17 10f x x x x 8.

3 2( ) 2 3 8 12f x x x x

9. 3 2( ) 6 11 3 2f x x x x 10.

4 3 2( ) 2 13 19 10 24f x x x x x

11. 4 3 2( ) 8 2 8 78 20f x x x x x 12.

4 3 2( ) 36 24 41 6 8f x x x x x

Foundations of Algebra 2 Name:

Unit 5 – Day 9 – Factoring Polynomials Equations Period: Date:

Find all of the possible roots of the following polynomials.

1. 3 2( ) 2 21 18f x x x x

Factors of the constant term:

Factors of the leading coefficient:

Possible rational zeros:

Simplified list of zeros:

2. 4 2( ) 13 36f x x x

Factors of the constant term:

Factors of the leading coefficient:

Possible rational zeros:

Simplified list of zeros:

Find all of the possible roots of the following polynomials.

3. 3 2( ) 11 34 104f x x x x

Factors of the constant term:

Factors of the leading coefficient:

Possible rational zeros:

Simplified list of zeros:

4. 4 2( ) 9 4 12f x x x x

Factors of the constant term:

Factors of the leading coefficient:

Possible rational zeros:

Simplified list of zeros:

Foundations of Algebra 2 Name:

Unit 5 – Day 10 – Apply the Rational Zero Theorem Period: Date:

Find all of the possible roots of the following polynomials.

1. 3 2( ) 3 19 4 12f x x x x

Factors of the constant term:

Factors of the leading coefficient:

Possible rational zeros:

Simplified list of zeros:

2. 3 2( ) 2 5 12 14f x x x x

Factors of the constant term:

Factors of the leading coefficient:

Possible rational zeros:

Simplified list of zeros:

Find all of the possible roots of the following polynomials.

3. 3 2( ) 2f x x x x

Factors of the constant term:

Factors of the leading coefficient:

Possible rational zeros:

Simplified list of zeros:

4. 3 2( ) 3 2 61 20f x x x x

Factors of the constant term:

Factors of the leading coefficient:

Possible rational zeros:

Simplified list of zeros:

Foundations of Algebra 2 Name:

Unit 5 – Day 11 – Find the Real Zeros Period: Date:

Using the Graph, find all the Real Zeros

1. 2. 3.

4. 5. 6.

Using a Calculator, find all the zeros.

7. 3 2( ) 2 21 18f x x x x 8.

4 3 2( ) 2 13 19 10 24f x x x x x

9. 3 2( ) 6 11 3 2f x x x x

Find the number of solutions or zeroes for each equation or function. Then look at the graph on a

calculator decide how many real solutions and imaginary solutions there are.

10. 3 2( ) 2 2 3f x x x x 11.

4 3 2( ) 8 2 8 78 20f x x x x x

Number of Solutions: __________ Number of Solutions: __________

Number of Real Solutions: __________ Number of Real Solutions: __________

Number of Imaginary Solutions: __________ Number of Imaginary Solutions: __________

Find all the zeroes for each function below.

12. 4 3 2( ) 36 24 41 6 8f x x x x x 13.

3 2( ) 7 15 9f x x x x

Number of Solutions: __________ Number of Solutions: __________

Number of Real Solutions: __________ Number of Real Solutions: __________

Number of Imaginary Solutions: __________ Number of Imaginary Solutions: __________