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Polynomial Power Review

A comprehensive review lesson covering polynomial addition, subtraction, multiplication, division, geometric applications, and multi-step factoring.

MRMahuya Roy Chowdhury

Polynomial Review Worksheet

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Algebraic Reasoning Unit 2 Review – Polynomials

AR.4A Question 1 Practice (Addition)

1A. The functions \( f(x) = x^2 + 13 \) and \( g(x) = 6x \). Find \( h(x) = (f + g)(x) \).

\( h(x) = \)

1B. The functions \( f(x) = 2x^2 - 8 \) and \( g(x) = 5x + 3 \). Find \( h(x) = (f + g)(x) \).

\( h(x) = \)

AR.4A Question 2 Practice (Table & Equation)

2A. Complete the table for \( h(x) = f(x) - g(x) \), then find the equation.

\( x \)	\( f(x) = 2x + 9 \)	\( g(x) = 6x - 1 \)
1	11	5
2	13	11
3	15	17
4	17	23
5	19	29

\( h(x) = \)

2B. Complete the table for \( h(x) = f(x) - g(x) \), then find the equation.

\( x \)	\( f(x) = 4x + 3 \)	\( g(x) = 7x - 2 \)
1	7	5
2	11	12
3	15	19
4	19	26
5	23	33

\( h(x) = \)

AR.4A Question 3 Practice (Subtract Polynomials)

3A. If \( h(x) = 4x - 60 \) and \( k(x) = 3x^2 - 5x + 40 \), which represents \( j(x) = h(x) - k(x) \)?

A. \( j(x) = 3x^2 - x - 20 \)

B. \( j(x) = -3x^2 + 9x - 100 \)

C. \( j(x) = 3x^2 + 9x - 20 \)

D. \( j(x) = -3x^2 - x + 100 \)

3B. If \( h(x) = 5x - 45 \) and \( k(x) = x^2 - 2x + 35 \), which represents \( j(x) = h(x) - k(x) \)?

A. \( j(x) = x^2 + 3x - 10 \)

B. \( j(x) = -x^2 + 7x - 80 \)

C. \( j(x) = -x^2 + 3x - 80 \)

D. \( j(x) = x^2 - 7x + 80 \)

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Algebraic Reasoning Unit 2 Review Page 2 of 4

AR.4A Question 4 Practice (Multiply Polynomials)

4A. If \( p(x) = 5x - 6 \) and \( q(x) = x^2 + 7 \), what is \( r(x) = (p \cdot q)(x) \)?

A. \( r(x) = 5x^3 - 42 \)

B. \( r(x) = 5x^2 + 29x - 42 \)

C. \( r(x) = 5x^3 - 6x^2 + 35x - 42 \)

D. \( r(x) = 6x^3 - 5x^2 + 35x - 42 \)

4B. If \( p(x) = 4x - 7 \) and \( q(x) = x^2 + 5 \), what is \( r(x) = (p \cdot q)(x) \)?

A. \( r(x) = 4x^3 - 7x^2 + 20x - 35 \)

B. \( r(x) = 4x^3 - 35 \)

C. \( r(x) = 4x^2 + 13x - 35 \)

D. \( r(x) = 5x^3 - 7x^2 + 20x - 35 \)

AR.4A Question 5 Practice (Cylinder Surface Area)

5A. 3x x

Radius is \( 3x \), height is \( x \). \( B(x) = 28.26x^2 \) and \( L(x) = 18.84x^2 \). Find \( S(x) = 2B(x) + L(x) \):

A. \( S(x) = 75.36x^2 \)

B. \( S(x) = 47.10x^2 \)

C. \( S(x) = 56.52x^2 \)

D. \( S(x) = 84.78x^2 \)

5B. x x

Radius is \( x \), height is \( x \). \( B(x) = 3.14x^2 \) and \( L(x) = 6.28x^2 \). Find \( S(x) = 2B(x) + L(x) \):

A. \( S(x) = 9.42x^2 \)

B. \( S(x) = 12.56x^2 \)

C. \( S(x) = 15.70x^2 \)

D. \( S(x) = 6.28x^2 \)

AR.4C Question 6 Practice (Polynomial Division)

6A. Find \( h(x) = f(x) \div g(x) \) for \( f(x) = 2x^2 + 7x - 15 \) and \( g(x) = 2x - 3 \).

\( h(x) = \)

6B. Find \( h(x) = f(x) \div g(x) \) for \( f(x) = 3x^2 - 11x - 4 \) and \( g(x) = 3x + 1 \).

\( h(x) = \)

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Algebraic Reasoning Unit 2 Review Page 3 of 4

AR.4C Question 7 Practice (Prism Height Division)

7A. Volume \( V(x) = 6x^3 + 19x^2 - 2x - 5 \) and base area \( B(x) = 2x + 1 \). Which function represents height \( h(x) \)?

A. \( h(x) = 3x - 5 \)

B. \( h(x) = 3x^2 - 5 \)

C. \( h(x) = 3x^2 - 8x + 5 \)

D. \( h(x) = 3x^2 + 8x - 5 \)

7B. Volume \( V(x) = 6x^3 + 17x^2 + 2x - 15 \) and base area \( B(x) = 2x + 3 \). Which function represents height \( h(x) \)?

A. \( h(x) = 3x^2 + 4x - 5 \)

B. \( h(x) = 3x^2 - 5 \)

C. \( h(x) = 3x^2 - 4x + 5 \)

D. \( h(x) = 3x - 5 \)

AR.4A Question 8 Practice (Prism Volume: \( V = l \times w \times h \))

8A. x + 5 ft x + 3 ft x ft

Write a function \( V \) for volume in cu ft (\( V = l \times w \times h \)):

\( V = \)

8B. x + 6 ft x + 1 ft x ft

Write a function \( V \) for volume in cu ft (\( V = l \times w \times h \)):

\( V = \)

AR.4D Question 9 Practice (Factor \( x^2 + bx + c \))

9A. Write as a product of factors:

\( g(x) = x^2 - 11x + 24 \)

\( g(x) = \) ( _______ )( _______ )

9B. Write as a product of factors:

\( g(x) = x^2 - 9x + 20 \)

\( g(x) = \) ( _______ )( _______ )

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Algebraic Reasoning Unit 2 Review Page 4 of 4

AR.4D Question 10 Practice (Factor \( ax^2 + bx + c \))

10A. Write as a product of factors:

\( g(x) = 2x^2 + 13x + 15 \)

\( g(x) = \) ( _______ )( _______ )

10B. Write as a product of factors:

\( g(x) = 3x^2 + 14x + 8 \)

\( g(x) = \) ( _______ )( _______ )

AR.4D Question 11 Practice (Factor GCF & Trinomial)

11A. Write as a product of factors:

\( g(x) = 3x^3 - 14x^2 + 8x \)

\( g(x) = \) ___( _______ )( _______ )

11B. Write as a product of factors:

\( g(x) = 2x^3 - 11x^2 + 12x \)

\( g(x) = \) ___( _______ )( _______ )

AR.4D Question 12 Practice (Factor by Grouping)

12A. Factor:

\( 6x^3 - 15x^2 + 8x - 20 \)

( ______________ )( ______________ )

12B. Factor:

\( 4x^3 - 6x^2 + 10x - 15 \)

( ______________ )( ______________ )

Page 4 of 4

Polynomial Review Answer Key

TEACHER RESOURCE • ANSWER KEY & SCORING GUIDE UNIT 2: POLYNOMIALS

Algebraic Reasoning Unit 2 Review – Answer Key

Total Items: 24 questions (12 pairs)

Target TEKS: AR.4A, AR.4C, AR.4D

Scoring: 1 pt per blank/MC (2 pts for Q2 tables)

#	Answer	TEKS	Type
1A	\( x^2 + 6x + 13 \)	AR.4A	Eq Edit
1B	\( 2x^2 + 5x - 5 \)	AR.4A	Eq Edit
2A
\( \{6, 2, -2, -6, -10\} \)

\( h(x) = -4x + 10 \)

| AR.4A | Table/Eq | | 2B |

\( \{2, -1, -4, -7, -10\} \)

\( h(x) = -3x + 5 \)

| AR.4A | Table/Eq | | 3A | B (\( -3x^2 + 9x - 100 \)) | AR.4A | MC | | 3B | B (\( -x^2 + 7x - 80 \)) | AR.4A | MC | | 4A | C (\( 5x^3 - 6x^2 + 35x - 42 \)) | AR.4A | MC | | 4B | A (\( 4x^3 - 7x^2 + 20x - 35 \)) | AR.4A | MC | | 5A | A (\( 75.36x^2 \)) | AR.4A | MC | | 5B | B (\( 12.56x^2 \)) | AR.4A | MC | | 6A | \( x + 5 \) | AR.4C | Eq Edit | | 6B | \( x - 4 \) | AR.4C | Eq Edit |

#	Answer	TEKS	Type
7A	D (\( 3x^2 + 8x - 5 \))	AR.4C	MC
7B	A (\( 3x^2 + 4x - 5 \))	AR.4C	MC
8A	\( x^3 + 8x^2 + 15x \)	AR.4A	Eq Edit
8B	\( x^3 + 7x^2 + 6x \)	AR.4A	Eq Edit
9A	\( (x - 3)(x - 8) \)	AR.4D	Eq Edit
9B	\( (x - 4)(x - 5) \)	AR.4D	Eq Edit
10A	\( (2x + 3)(x + 5) \)	AR.4D	Eq Edit
10B	\( (3x + 2)(x + 4) \)	AR.4D	Eq Edit
11A	\( x(3x - 2)(x - 4) \)	AR.4D	Eq Edit
11B	\( x(2x - 3)(x - 4) \)	AR.4D	Eq Edit
12A	\( (2x - 5)(3x^2 + 4) \)

Instructional & Grading Notes

Q2 Tables: Award 1 point for the correct column entries and 1 point for the simplified linear equation.
Q3 Subtraction: Watch for negative sign distribution errors: \( -(3x^2 - 5x + 40) = -3x^2 + 5x - 40 \).
Q5 Surface Area: Remind students to double base area: \( S(x) = 2B(x) + L(x) \).
Q11 GCF: Emphasize factoring out \( x \) first before factoring the remaining quadratic trinomial.

Page 1 of 2 • Teacher Answer Key

Worked Solutions & Factoring Guide Page 2 of 2

Polynomial Operations (Addition, Subtraction, Multiplication)

1A: \( (x^2 + 13) + (6x) = x^2 + 6x + 13 \)
3A: \( (4x - 60) - (3x^2 - 5x + 40) = 4x - 60 - 3x^2 + 5x - 40 = -3x^2 + 9x - 100 \)
4A: \( (5x - 6)(x^2 + 7) = 5x(x^2 + 7) - 6(x^2 + 7) = 5x^3 + 35x - 6x^2 - 42 = 5x^3 - 6x^2 + 35x - 42 \)

1B: \( (2x^2 - 8) + (5x + 3) = 2x^2 + 5x - 5 \)
3B: \( (5x - 45) - (x^2 - 2x + 35) = 5x - 45 - x^2 + 2x - 35 = -x^2 + 7x - 80 \)
4B: \( (4x - 7)(x^2 + 5) = 4x(x^2 + 5) - 7(x^2 + 5) = 4x^3 + 20x - 7x^2 - 35 = 4x^3 - 7x^2 + 20x - 35 \)

Geometric Models & Polynomial Division

5A: \( S(x) = 2(28.26x^2) + 18.84x^2 = 56.52x^2 + 18.84x^2 = 75.36x^2 \)
6A: Factor numerator: \( 2x^2 + 7x - 15 = (2x - 3)(x + 5) \). Divide by \( (2x - 3) \implies x + 5 \)
7A: Divide \( \frac{6x^3 + 19x^2 - 2x - 5}{2x + 1} = 3x^2 + 8x - 5 \) (Choice D)
8A: \( V = x(x + 3)(x + 5) = x(x^2 + 8x + 15) = x^3 + 8x^2 + 15x \)

5B: \( S(x) = 2(3.14x^2) + 6.28x^2 = 6.28x^2 + 6.28x^2 = 12.56x^2 \)
6B: Factor numerator: \( 3x^2 - 11x - 4 = (3x + 1)(x - 4) \). Divide by \( (3x + 1) \implies x - 4 \)
7B: Divide \( \frac{6x^3 + 17x^2 + 2x - 15}{2x + 3} = 3x^2 + 4x - 5 \) (Choice A)
8B: \( V = x(x + 1)(x + 6) = x(x^2 + 7x + 6) = x^3 + 7x^2 + 6x \)

Factoring Strategies (Trinomials, GCF, and Grouping)

Q9 (Trinomials \( a = 1 \)): Find factors of \( c \) adding to \( b \).
• 9A: \( x^2 - 11x + 24 \implies (-3)(-8) = 24 \text{ and } (-3) + (-8) = -11 \implies \mathbf{(x - 3)(x - 8)} \)
• 9B: \( x^2 - 9x + 20 \implies (-4)(-5) = 20 \text{ and } (-4) + (-5) = -9 \implies \mathbf{(x - 4)(x - 5)} \)

Q10 (Trinomials \( a > 1 \)): Split middle term or box method.
• 10A: \( 2x^2 + 13x + 15 \implies ac = 30 \); factors \( 10, 3 \implies 2x^2 + 10x + 3x + 15 = 2x(x + 5) + 3(x + 5) = \mathbf{(2x + 3)(x + 5)} \)
• 10B: \( 3x^2 + 14x + 8 \implies ac = 24 \); factors \( 12, 2 \implies 3x^2 + 12x + 2x + 8 = 3x(x + 4) + 2(x + 4) = \mathbf{(3x + 2)(x + 4)} \)

Q11 (GCF + Trinomial): Pull out greatest common factor \( x \) first.
• 11A: \( 3x^3 - 14x^2 + 8x = x(3x^2 - 14x + 8) \); factors of \( 24 \) adding to \( -14 \) are \( -12, -2 \implies \mathbf{x(3x - 2)(x - 4)} \)
• 11B: \( 2x^3 - 11x^2 + 12x = x(2x^2 - 11x + 12) \); factors of \( 24 \) adding to \( -11 \) are \( -8, -3 \implies \mathbf{x(2x - 3)(x - 4)} \)

Q12 (Grouping 4 Terms): Group first two and last two terms.
• 12A: \( (6x^3 - 15x^2) + (8x - 20) = 3x^2(2x - 5) + 4(2x - 5) = \mathbf{(2x - 5)(3x^2 + 4)} \)
• 12B: \( (4x^3 - 6x^2) + (10x - 15) = 2x^2(2x - 3) + 5(2x - 3) = \mathbf{(2x - 3)(2x^2 + 5)} \)

Page 2 of 2 • Teacher Answer Key

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