Rational Expressions Test Answer Key with Scoring – Algebra 2

Question 1 Simplify with restrictions (2 points)

(x − 7)(x + 3)(x − 7)(x + 7) = x + 3x + 7, with x ≠ 7 and x ≠ −7.

Scoring: 1 point for the simplified expression, 1 point for both excluded values.

Question 2 Divide and simplify (3 points)

Factor: (x − 2)(x − 4)x(x + 1) · x + 1x − 2.

Cancel x − 2 and x + 1: x − 4x.

Scoring: 1 point for the reciprocal, 1 for factoring, 1 for the simplified answer.

Question 3 Subtract rational expressions (3 points)

LCD = (x + 4)(x − 6). 5(x − 6) − 3(x + 4)(x + 4)(x − 6) = 5x − 30 − 3x − 12(x + 4)(x − 6).

2x − 42(x + 4)(x − 6) or 2(x − 21)(x + 4)(x − 6)

Scoring: 1 point for the LCD, 1 for the numerators, 1 for the result.

Question 4 Solve a rational equation (3 points)

Multiply by x(x + 6): 2(x + 6) + 5x = x(x + 6), so 7x + 12 = x² + 6x and x² − x − 12 = 0.

(x − 4)(x + 3) = 0 gives x = 4 or x = −3. Neither is 0 or −6.

x = 4 or x = −3

Scoring: 1 point for the cleared equation, 1 for solving, 1 for the check.

Question 5 Extraneous solution (2 points)

Multiply by x − 6: x = 3(x − 6) + 6 = 3x − 12, so x = 6.

x = 6 makes the denominator 0, so it is extraneous.

No solution

Scoring: 1 point for the candidate x = 6, 1 point for rejecting it.

Question 6 Working together (3 points)

Rosa: 112 per hour; Theo: 120 per hour. Together 560 + 360 = 860 = 215 per hour.

Time = 152 = 7.5 hours.

7.5 hours

Scoring: 1 point for the rates, 1 for the sum, 1 for the time.

Question 7 Solutions from a graph (2 points)

a) The graphs cross at (2, 4) and (−4, −2), so x = 2 and x = −4.

b) For x = 2: 8/2 = 4 and 2 + 2 = 4 ✓.

Scoring: 1 point each part.

Question 8 Writing from excluded values (2 points)

The denominator must be 0 exactly at x = −3 and x = 4, so it has factors x + 3 and x − 4.

1(x + 3)(x − 4)

Scoring: 1 point for each correct factor.