Adding, Subtracting and Multiplying Polynomials: Algebra 1 Math Test Answer Key – PDF Download

Question 1: Standard form and degree (2 points)

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

Scoring: 1 point for the correct order with correct signs; 1 point for both the degree and the leading coefficient.

Question 2: Add and subtract (2 points)

1. \(4 x^{2} + 3 x + 6\)
2. Distribute the negative: \(3x^{3}+5x-8-x^{3}+2x^{2}-5x-1\). The \(x\) terms cancel. Result: \(2 x^{3} + 2 x^{2} – 9\).

Scoring: 1 point each part. In part (b), award the point only if the sign of every term of the second polynomial was changed.

Question 3: Subtract with two variables (2 points)

\(7a^{2}b-3ab+4b^{2}-2a^{2}b-ab+6b^{2}\)

Combine like terms: \(5a^{2}b-4ab+10b^{2}\).

\(5 a^{2} b – 4 a b + 10 b^{2}\)

Scoring: 1 point for distributing the negative sign correctly; 1 point for the combined result.

Question 4: Multiply polynomials (2 points)

1. \(-12x^{3}+8x^{2}-24x\)
2. \(x^{3}+4x^{2}-2x-6x^{2}-24x+12\), which combines to \(x^{3} – 2 x^{2} – 26 x + 12\).

Scoring: 1 point each part. Part (b) needs all six products and correct combining.

Question 5: Multiply two binomials (2 points)

Cells: \(3x^{2}\), \(4x\), \(-15x\), \(-20\). Combine \(4x-15x=-11x\).

\(3 x^{2} – 11 x – 20\)

Scoring: 1 point for the four correct cells; 1 point for combining like terms into the answer.

Question 6: Special products (2 points)

1. \((4x)^{2}-2(4x)(3)+3^{2}=16x^{2}-24x+9\)
2. \((6x)^{2}-5^{2}=36x^{2}-25\)

Scoring: 1 point each part. No credit for \(16x^{2}+9\) in part (a).

Question 7: Community garden (3 points)

1. \((2x+3)(x+5)=2x^{2}+10x+3x+15=2x^{2}+13x+15\) square feet.
2. \(2(36)+13(6)+15=72+78+15=165\). Check: \(15\times 11=165\) square feet.
3. \(2[(2x+3)+(x+5)]=2(3x+8)=6x+16\) feet.

Scoring: 1 point per part.

Question 8: Bakery profit (3 points)

1. \(P=(-x^{2}+60x)-(12x+150)=-x^{2}+60x-12x-150\), so \(P=- x^{2} + 48 x – 150\).
2. \(-100+480-150=230\). Check: \(500-270=230\). The profit is \$230.

Scoring: part (a) 1 point for subtracting the whole cost (distributing the negative) and 1 point for the simplified result; part (b) 1 point.

Question 9: Explain the error (2 points)

1. Left side: \((1+4)^{2}=25\). Liam’s right side: \(1+16=17\). Since \(25\neq 17\), the result is wrong.
2. \(x^{2} + 8 x + 16\). The square has parts \(x^{2}\), \(4x\), \(4x\) and \(16\).

Scoring: 1 point for a correct numerical check; 1 point for the correct expansion.

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