Algebra 2 Polynomial Operations and Identities Test Solutions and Scoring (PDF)

1. \(5x^{2} + 2x + 5\)

2. \(3x^{2} – 8x + 9\)

(a) \(5x^{2} + 2x + 5\); (b) \(3x^{2} – 8x + 9\)

Scoring: 1 point for each part.

1. \(x^{2} + 5x – 24\)

2. \(2x^{3} – 8x^{2} + 6x + x^{2} – 4x + 3\)

   \(2x^{3} – 7x^{2} + 2x + 3\)

(a) \(x^{2} + 5x – 24\); (b) \(2x^{3} – 7x^{2} + 2x + 3\)

Scoring: Part (a): 1 point. Part (b): 1 point for a correct distribution, 1 point for combining like terms.

1. \(16x^{2} – 8x + 1\)

2. \(25x^{2} – 36\)

(a) \(16x^{2} – 8x + 1\); (b) \(25x^{2} – 36\)

Scoring: 1 point for each part.

\((x + 6)(2x + 1) = 2x^{2} + x + 12x + 6\).

\(2x^{2} + 13x + 6\)

Scoring: 1 point for the four partial products, 1 point for the combined polynomial.

\(98^{2} = (100 – 2)^{2} = 10000 – 400 + 4 = 9604\).

\(61 \cdot 59 = (60 + 1)(60 – 1) = 3600 – 1 = 3599\).

9,604 and 3,599

Scoring: 1 point for each computation.

Legs: \(x^{2} – y^{2} = 25 – 4 = 21\) and \(2xy = 20\). Hypotenuse: \(x^{2} + y^{2} = 29\).

Check: \(21^{2} + 20^{2} = 441 + 400 = 841 = 29^{2}\).

21, 20, 29

Scoring: 1 point for each of the three sides generated, 1 point for the verification (3 points total: sides 2, check 1).

Outer: \((x + 4)(x + 7) = x^{2} + 11x + 28\). Photo: \(x(x + 3) = x^{2} + 3x\).

Subtract.

\(8x + 28\) square inches

Scoring: 1 point for the outer and inner areas, 1 point for the difference.

The middle term \(2(3x)(-2) = -12x\) is missing, and the sign pattern for a square of a difference is \(a^{2} – 2ab + b^{2}\).

\(9x^{2} – 12x + 4\)

Scoring: 1 point for the error, 1 point for the correct expansion.

The degree is \(2 + 3 = 5\). The leading coefficient is \(3 \cdot 1 = 3\).

Degree 5, leading coefficient 3

Scoring: 1 point for the degree, 1 point for the leading coefficient.