Factoring Polynomials: Algebra 1 Math Test Answer Key – PDF Download

Algebra 1 Factoring Test Answer Key: questions 1–5

Question 1: Greatest common factor (2 points)

1. \(5 x^{2} (3 x – 2)\)
2. \(6 a b (2 a + 3 b)\)

Scoring: 1 point each part. Award no point if the GCF is only partly removed, such as \(5(3x^{3}-2x^{2})\).

Question 2: Factor trinomials (2 points)

1. Product 24, sum 11: 3 and 8. \((x + 3) (x + 8)\)
2. Product \(-21\), sum \(-4\): \(-7\) and 3. \((x – 7) (x + 3)\)

Scoring: 1 point each part for the correct factors with correct signs.

Question 3: Factor with a leading coefficient (2 points)

1. \(ac=24\), numbers 4 and 6: \(3x^{2}+4x+6x+8=x(3x+4)+2(3x+4)\). \((3 x + 4) (x + 2)\)
2. \(ac=-20\), numbers 2 and \(-10\): \(4x^{2}+2x-10x-5=2x(2x+1)-5(2x+1)\). \((2 x – 5) (2 x + 1)\)

Scoring: 1 point each part. Accept the factors in either order.

Question 4: Special patterns (2 points)

1. \((7x)^{2}-6^{2}\): \((7 x – 6) (7 x + 6)\)
2. \((3x)^{2}+2(3x)(5)+5^{2}\): \((3 x + 5)^{2}\)

Scoring: 1 point each part.

Question 5: Factor by grouping (2 points)

1. \(x^{2}(x+5)+2(x+5)\). \((x^{2} + 2) (x + 5)\)
2. \(3x^{2}(x-2)-4(x-2)\). \((3 x^{2} – 4) (x – 2)\)

Scoring: 1 point each part. In part (b), the sign of the second group must be changed to factor out \(-4\).

Algebra 1 Factoring Test Answer Key: questions 6–9

Question 6: Factor completely (3 points)

1. \(3(x^{2}-16)\): \(3 (x – 4) (x + 4)\)
2. \(2x(x^{2}+6x+9)\): \(2 x (x + 3)^{2}\)
3. \(ac=-24\), numbers 3 and \(-8\): \(6x^{2}+3x-8x-4=3x(2x+1)-4(2x+1)\). \((3 x – 4) (2 x + 1)\)

Scoring: 1 point each part. Part (a) needs both the GCF and the difference of squares.

Question 7: Zeros and the graph (3 points)

1. Product \(-6\), sum \(-1\): \(-3\) and 2. \((x – 3) (x + 2)\)
2. \(x-3=0\) or \(x+2=0\), so \(x=3\) or \(x=-2\).
3. The x-intercepts are \((-2,0)\) and \((3,0)\).

Scoring: 1 point for each part. In part (c), both coordinates must be given as ordered pairs.

Question 8: Poster dimensions (2 points)

1. \(ac=24\), numbers 3 and 8: \(2x^{2}+3x+8x+12=x(2x+3)+4(2x+3)\). \((2 x + 3) (x + 4)\). The length is \(2x+3\) inches.
2. Length \(2(5)+3=13\) inches, width \(5+4=9\) inches. Check: \(13\times 9=117\) and \(50+55+12=117\).

Scoring: part (a) 1 point; part (b) 1 point for both values.

Question 9: Is it complete? (2 points)

1. Each binomial has a common factor of 2: \(2x-10=2(x-5)\) and \(2x+10=2(x+5)\), so further factoring is possible.
2. \(4(x^{2}-25)\) gives \(4 (x – 5) (x + 5)\).

Scoring: 1 point for naming the remaining common factor; 1 point for the complete factorization.

Scoring summary

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Algebra 1 Factoring Polynomials test answer key

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