Algebra 1 Factoring Polynomials Test Solutions and Scoring (PDF)

Algebra 1 math test solutions: Factoring Polynomials – Maths PDF

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This factoring test answers for Algebra 1 covers factoring polynomials (Common Core A-SSE.A.2, A-SSE.B.3a, A-APR.B.3), with a free PDF to print.

Grade the factoring test with this key. Every question lists the points available and how to split them between method and result. Students who show clear work should receive partial credit, even if the final answer is not correct. The test totals 20 points in all.

Algebra 1 · Common Core State Standards: A-SSE.A.2, A-SSE.B.3a, A-APR.B.3

Factoring test answers: worked solutions

Question 1 – Greatest common factor [2 pts]

The GCF is \(8x^{2}\): \(\frac{24x^{3}}{8x^{2}} = 3x\) and \(\frac{16x^{2}}{8x^{2}} = 2\).

\(8x^{2}(3x + 2)\)

Scoring: 1 point for the GCF; 1 point for the remaining factor.

Question 2 – Simple trinomial [2 pts]

Cells \(8x\) and \(3x\) add to \(11x\).

Numbers with product 24 and sum 11: 3 and 8.

\((x + 3)(x + 8)\)

Scoring: 1 point for the pair 3 and 8; 1 point for the factors.

Question 3 – Mixed signs [2 pts]

Product \(-32\) and sum \(-4\): \(-8\) and 4.

\((x – 8)(x + 4)\)

Scoring: 1 point for the pair; 1 point for the factors.

Question 4 – Leading coefficient 3 [2 pts]

Multiply \(3 \cdot (-6) = -18\). The numbers 9 and \(-2\) add to 7.

Split: \(3x^{2} + 9x – 2x – 6 = 3x(x + 3) – 2(x + 3)\).

\((3x – 2)(x + 3)\)

Scoring: 1 point for splitting the middle term; 1 point for the factors.

Question 5 – Difference of squares [2 pts]

\(x^{2} – 9^{2} = (x – 9)(x + 9)\).

\((x – 9)(x + 9)\)

Scoring: 1 point for recognizing the pattern; 1 point for the factors.

Question 6 – Perfect square trinomial [2 pts]

\(49 = 7^{2}\) and \(2(7) = 14\), so it is a perfect square.

\((x – 7)^{2}\)

Scoring: 1 point for the pattern; 1 point for the factored form.

Question 7 – Factor completely and find zeros [3 pts]

a) Factor out 3: \(3(x^{2} – 16)\). Then \(x^{2} – 16 = (x – 4)(x + 4)\).

b) Set \(x – 4 = 0\) or \(x + 4 = 0\).

\(3(x – 4)(x + 4)\); \(x = 4\) or \(x = -4\)

Scoring: part a, 1 point for the GCF and 1 point for the difference of squares; part b, 1 point.

Question 8 – Rectangle dimensions [3 pts]

Length \(x + 8\).

a) Product 16 and sum 10: 2 and 8, so \(x^{2} + 10x + 16 = (x + 2)(x + 8)\).

b) The width is \(x + 2\), so the length is \(x + 8\).

\((x + 2)(x + 8)\); length \(x + 8\) m

Scoring: part a, 1 point for the pair and 1 point for the factors; part b, 1 point.

Question 9 – Ball landing time [2 pts]

Factor out \(-16t\): \(h(t) = -16t(t – 3)\).

The height is 0 when \(t = 0\) or \(t = 3\). The ball lands at \(t = 3\).

\(-16t(t – 3)\); 3 seconds

Scoring: 1 point for the factored form; 1 point for the landing time.

Total score

Total

20 points

Key facts behind this factoring test answers

Key facts

Factor out the greatest common factor (GCF) first.
For x² + bx + c, find two numbers with product c and sum b.
Difference of squares: a² – b² = (a + b)(a – b).
Perfect square trinomials: a² ± 2ab + b² = (a ± b)².
If a product is zero, one of its factors is zero, so factored form shows the zeros.

How to use this factoring test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Algebra 1 chapters: Algebra 1 math.

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