Algebra 2 Rational Expressions Test Solutions and Scoring (PDF)

\(x^{2} – 3x – 10 = (x – 5)(x + 2)\), so \(x \ne 5, -2\).

\(\frac{x – 5}{(x – 5)(x + 2)} = \frac{1}{x + 2}\).

\(x \ne 5, -2\); \(\frac{1}{x + 2}\)

Scoring: 1 point for the excluded values, 1 point for the simplified form.

\(\frac{(x + 5)(x – 3)}{(x – 3)(x + 3)}\).

\(\frac{x + 5}{x + 3}\)

Scoring: 1 point for factoring, 1 point for the result.

\(\frac{2x}{x + 4} \cdot \frac{(x – 4)(x + 4)}{6x} = \frac{2(x – 4)}{6}\).

\(\frac{x – 4}{3}\)

Scoring: 1 point for factoring, 1 point for the result.

\(\frac{(x – 2)(x + 2)}{x + 3} \cdot \frac{(x + 3)^{2}}{x + 2} = (x – 2)(x + 3)\).

\((x – 2)(x + 3)\)

Scoring: 1 point for the reciprocal, 1 point for the simplified product.

LCD \((x – 3)(x + 1)\): \(\frac{2(x + 1) + 4(x – 3)}{(x – 3)(x + 1)} = \frac{2x + 2 + 4x – 12}{(x – 3)(x + 1)}\).

\(\frac{6x – 10}{(x – 3)(x + 1)}\)

Scoring: 1 point for the common denominator, 1 point for the numerators, 1 point for the simplified result.

Multiply the top and bottom by \(3x\): \(\frac{3 + x}{3 – x}\).

\(\frac{3 + x}{3 – x}\)

Scoring: 1 point for multiplying by the LCD, 1 point for the result.

\(\frac{1}{6} + \frac{1}{4} = \frac{5}{12}\) of the tank per hour, so \(t = \frac{12}{5}\) hours.

2.4 hours

Scoring: 1 point for the combined rate, 1 point for the time.

\(x^{4} – 16 = (x^{2} – 4)(x^{2} + 4)\), so \(g(x) = x^{2} + 4\).

Excluded: \(x^{2} – 4 = 0\) gives \(x = \pm 2\).

\(g(x) = x^{2} + 4\); \(x \ne 2, -2\)

Scoring: 1 point for the factorization, 1 point for the simplified form, 1 point for the excluded values.

At \(x = 6\): \(\frac{12}{6} = 2\) but \(6 + 1 = 7\). The values differ, so the claim is false.

Correct: \(\frac{x + 6}{6} = \frac{x}{6} + 1\).

False; the correct form is x/6 + 1

Scoring: 1 point for the counterexample, 1 point for the correct form.