Polynomial Factoring Practice Worksheet
AR.4D Practice
Factoring Polynomials: GCF & Trinomials
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Date:
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AR.4D Question 11 Practice (Factor GCF & Trinomial)
11A. Write as a product of factors:
\(g(x) = 3x^3 - 14x^2 + 8x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
11B. Write as a product of factors:
\(g(x) = 2x^3 - 11x^2 + 12x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
AR.4D Question 12 Practice (Leading Coefficient 5)
12A. Write as a product of factors:
\(g(x) = 5x^3 - 17x^2 + 6x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
12B. Write as a product of factors:
\(g(x) = 5x^3 - 23x^2 + 12x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
Unit 4: Polynomial Functions • Standard AR.4D Page 1 (Sheet 1 Front)
AR.4D Practice
Factoring Polynomials: Continued
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AR.4D Question 13 Practice (Leading Coefficient 4)
13A. Write as a product of factors:
\(g(x) = 4x^3 - 19x^2 + 12x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
13B. Write as a product of factors:
\(g(x) = 4x^3 - 11x^2 + 6x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
AR.4D Question 14 Practice (Leading Coefficient 6)
14A. Write as a product of factors:
\(g(x) = 6x^3 - 19x^2 + 10x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
14B. Write as a product of factors:
\(g(x) = 6x^3 - 17x^2 + 12x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
Unit 4: Polynomial Functions • Standard AR.4D Page 2 (Sheet 1 Back)
AR.4D Practice
Factoring Polynomials: Mixed & Advanced
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Date:
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AR.4D Question 15 Practice (Mixed Signs Factoring)
15A. Write as a product of factors:
\(g(x) = 2x^3 + 7x^2 - 15x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
15B. Write as a product of factors:
\(g(x) = 3x^3 + 13x^2 - 10x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
AR.4D Question 16 Practice (Leading Coefficient 7)
16A. Write as a product of factors:
\(g(x) = 7x^3 - 22x^2 + 3x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
16B. Write as a product of factors:
\(g(x) = 7x^3 - 29x^2 + 4x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
Unit 4: Polynomial Functions • Standard AR.4D Page 3 (Sheet 2 Front)
AR.4D Practice
Factoring Polynomials: Higher Coefficients
Name:
Period:
AR.4D Question 17 Practice (Leading Coefficient 8)
17A. Write as a product of factors:
\(g(x) = 8x^3 - 18x^2 + 9x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
17B. Write as a product of factors:
\(g(x) = 8x^3 - 22x^2 + 15x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
AR.4D Question 18 Practice (Leading Coefficient 10)
18A. Write as a product of factors:
\(g(x) = 10x^3 - 29x^2 + 10x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
18B. Write as a product of factors:
\(g(x) = 10x^3 - 31x^2 + 15x\)
\(g(x) = \underline{\hspace{24px}}\;(\underline{\hspace{64px}})(\underline{\hspace{64px}})\)
Unit 4: Polynomial Functions • Standard AR.4D Page 4 (Sheet 2 Back)
Polynomial Factoring Answer Key
Teacher Reference
Answer Key: Problems 11 – 14
Unit 4: Polynomial Functions
Standard TEKS AR.4D
Part 1: Questions 11 & 12 Practice Sheet 1 Front
11A. \(g(x) = 3x^3 - 14x^2 + 8x\) Q11
Step 1 (GCF): \(x(3x^2 - 14x + 8)\)
Step 2 (ac = 24): \(-12 \times -2 = 24\), sum \(= -14\)
Step 3 (Group): \(3x(x - 4) - 2(x - 4)\)
\(g(x) = \mathbf{x(3x - 2)(x - 4)}\)
11B. \(g(x) = 2x^3 - 11x^2 + 12x\) Q11
Step 1 (GCF): \(x(2x^2 - 11x + 12)\)
Step 2 (ac = 24): \(-8 \times -3 = 24\), sum \(= -11\)
Step 3 (Group): \(2x(x - 4) - 3(x - 4)\)
\(g(x) = \mathbf{x(2x - 3)(x - 4)}\)
12A. \(g(x) = 5x^3 - 17x^2 + 6x\) Q12
Step 1 (GCF): \(x(5x^2 - 17x + 6)\)
Step 2 (ac = 30): \(-15 \times -2 = 30\), sum \(= -17\)
Step 3 (Group): \(5x(x - 3) - 2(x - 3)\)
\(g(x) = \mathbf{x(5x - 2)(x - 3)}\)
12B. \(g(x) = 5x^3 - 23x^2 + 12x\) Q12
Step 1 (GCF): \(x(5x^2 - 23x + 12)\)
Step 2 (ac = 60): \(-20 \times -3 = 60\), sum \(= -23\)
Step 3 (Group): \(5x(x - 4) - 3(x - 4)\)
\(g(x) = \mathbf{x(5x - 3)(x - 4)}\)
Part 2: Questions 13 & 14 Practice Sheet 1 Back
13A. \(g(x) = 4x^3 - 19x^2 + 12x\) Q13
Step 1 (GCF): \(x(4x^2 - 19x + 12)\)
Step 2 (ac = 48): \(-16 \times -3 = 48\), sum \(= -19\)
Step 3 (Group): \(4x(x - 4) - 3(x - 4)\)
\(g(x) = \mathbf{x(4x - 3)(x - 4)}\)
13B. \(g(x) = 4x^3 - 11x^2 + 6x\) Q13
Step 1 (GCF): \(x(4x^2 - 11x + 6)\)
Step 2 (ac = 24): \(-8 \times -3 = 24\), sum \(= -11\)
Step 3 (Group): \(4x(x - 2) - 3(x - 2)\)
\(g(x) = \mathbf{x(4x - 3)(x - 2)}\)
14A. \(g(x) = 6x^3 - 19x^2 + 10x\) Q14
Step 1 (GCF): \(x(6x^2 - 19x + 10)\)
Step 2 (ac = 60): \(-15 \times -4 = 60\), sum \(= -19\)
Step 3 (Group): \(3x(2x - 5) - 2(2x - 5)\)
\(g(x) = \mathbf{x(3x - 2)(2x - 5)}\)
14B. \(g(x) = 6x^3 - 17x^2 + 12x\) Q14
Step 1 (GCF): \(x(6x^2 - 17x + 12)\)
Step 2 (ac = 72): \(-9 \times -8 = 72\), sum \(= -17\)
Step 3 (Group): \(3x(2x - 3) - 4(2x - 3)\)
\(g(x) = \mathbf{x(3x - 4)(2x - 3)}\)
Grading Note & Common Student Pitfalls (Q11–Q14):
Students frequently forget to write the common factor \(x\) in the final line after correctly factoring the trinomial. Ensure full credit requires all three factors: \(g(x) = x(ax + b)(cx + d)\).
Unit 4 Answer Key • Standard AR.4D Page 1 (Problems 11 – 14)
Teacher Reference
Answer Key: Problems 15 – 18
Unit 4: Polynomial Functions
Standard TEKS AR.4D
Part 3: Questions 15 & 16 Practice Sheet 2 Front
15A. \(g(x) = 2x^3 + 7x^2 - 15x\) Q15
Step 1 (GCF): \(x(2x^2 + 7x - 15)\)
Step 2 (ac = -30): \(10 \times -3 = -30\), sum \(= 7\)
Step 3 (Group): \(2x(x + 5) - 3(x + 5)\)
\(g(x) = \mathbf{x(2x - 3)(x + 5)}\)
15B. \(g(x) = 3x^3 + 13x^2 - 10x\) Q15
Step 1 (GCF): \(x(3x^2 + 13x - 10)\)
Step 2 (ac = -30): \(15 \times -2 = -30\), sum \(= 13\)
Step 3 (Group): \(3x(x + 5) - 2(x + 5)\)
\(g(x) = \mathbf{x(3x - 2)(x + 5)}\)
16A. \(g(x) = 7x^3 - 22x^2 + 3x\) Q16
Step 1 (GCF): \(x(7x^2 - 22x + 3)\)
Step 2 (ac = 21): \(-21 \times -1 = 21\), sum \(= -22\)
Step 3 (Group): \(7x(x - 3) - 1(x - 3)\)
\(g(x) = \mathbf{x(7x - 1)(x - 3)}\)
16B. \(g(x) = 7x^3 - 29x^2 + 4x\) Q16
Step 1 (GCF): \(x(7x^2 - 29x + 4)\)
Step 2 (ac = 28): \(-28 \times -1 = 28\), sum \(= -29\)
Step 3 (Group): \(7x(x - 4) - 1(x - 4)\)
\(g(x) = \mathbf{x(7x - 1)(x - 4)}\)
Part 4: Questions 17 & 18 Practice Sheet 2 Back
17A. \(g(x) = 8x^3 - 18x^2 + 9x\) Q17
Step 1 (GCF): \(x(8x^2 - 18x + 9)\)
Step 2 (ac = 72): \(-12 \times -6 = 72\), sum \(= -18\)
Step 3 (Group): \(4x(2x - 3) - 3(2x - 3)\)
\(g(x) = \mathbf{x(4x - 3)(2x - 3)}\)
17B. \(g(x) = 8x^3 - 22x^2 + 15x\) Q17
Step 1 (GCF): \(x(8x^2 - 22x + 15)\)
Step 2 (ac = 120): \(-12 \times -10 = 120\), sum \(= -22\)
Step 3 (Group): \(4x(2x - 3) - 5(2x - 3)\)
\(g(x) = \mathbf{x(4x - 5)(2x - 3)}\)
18A. \(g(x) = 10x^3 - 29x^2 + 10x\) Q18
Step 1 (GCF): \(x(10x^2 - 29x + 10)\)
Step 2 (ac = 100): \(-25 \times -4 = 100\), sum \(= -29\)
Step 3 (Group): \(5x(2x - 5) - 2(2x - 5)\)
\(g(x) = \mathbf{x(5x - 2)(2x - 5)}\)
18B. \(g(x) = 10x^3 - 31x^2 + 15x\) Q18
Step 1 (GCF): \(x(10x^2 - 31x + 15)\)
Step 2 (ac = 150): \(-25 \times -6 = 150\), sum \(= -31\)
Step 3 (Group): \(5x(2x - 5) - 3(2x - 5)\)
\(g(x) = \mathbf{x(5x - 3)(2x - 5)}\)
Grading Note & Common Student Pitfalls (Q15–Q18):
In Question 15 (opposite signs), check that students assign the positive and negative signs to the correct binomial factors. In Questions 17 & 18, remind students that factor orders like \((2x - 3)(4x - 5)\) and \((4x - 5)(2x - 3)\) are equivalent.
Unit 4 Answer Key • Standard AR.4D Page 2 (Problems 15 – 18)
