Bluebook SAT Test Equivalent Expressions 6

Question 1

Which of the following is equivalent to (6x² + x − 2) / (3x² + 5x + 2), given that the expression is defined?

Accepted Answer

(2x − 1) / (x + 1)

Answer

Factor the numerator: 6x² + x − 2 = (2x − 1)(3x + 2). Factor the denominator: 3x² + 5x + 2 = (3x + 2)(x + 1). The common factor (3x + 2) cancels, leaving (2x − 1)/(x + 1).

Question 2

Which of the following is equivalent to 4x² − 20x + 19?

Accepted Answer

(2x − 5)² − 6

Answer

Notice that (2x − 5)² = 4x² − 20x + 25. Since the original expression is 4x² − 20x + 19, we subtract 6 from (2x − 5)²: (2x − 5)² − 6 = 4x² − 20x + 25 − 6 = 4x² − 20x + 19. Choice C incorrectly factors out 4 with a wrong inner term; choice D has an incorrect coefficient.

Question 3

Which expression is equivalent to (1/x − 1/y) / (1/x + 1/y), where x ≠ 0, y ≠ 0, and x ≠ −y?

Accepted Answer

(y − x) / (y + x)

Answer

Combine the numerator: (1/x − 1/y) = (y − x)/(xy). Combine the denominator: (1/x + 1/y) = (y + x)/(xy). Dividing gives [(y − x)/(xy)] ÷ [(y + x)/(xy)] = (y − x)/(y + x). Choice A, (x − y)/(x + y), has the opposite sign in the numerator — a common trap.

Question 4

Which of the following is equivalent to (x³ − 8) / (x² − 4), given that x ≠ 2 and x ≠ −2?

Accepted Answer

(x² + 2x + 4) / (x + 2)

Answer

Factor using difference of cubes: x³ − 8 = (x − 2)(x² + 2x + 4). Factor the denominator as a difference of squares: x² − 4 = (x − 2)(x + 2). Cancel the common (x − 2) factor to get (x² + 2x + 4)/(x + 2). Choice A uses the wrong trinomial (difference of cubes formula gives +2x, not −2x).

Question 5

Which expression is equivalent to (27x⁶ / y⁻³)^(2/3)?

Accepted Answer

9x⁴y²

Answer

First, rewrite y⁻³ in the denominator: dividing by y⁻³ equals multiplying by y³, so the base becomes 27x⁶y³. Apply the 2/3 exponent: 27^(2/3) = (3³)^(2/3) = 3² = 9; (x⁶)^(2/3) = x^4; (y³)^(2/3) = y². The result is 9x⁴y². A common error is mishandling the negative exponent in the denominator and getting y in the denominator instead.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Equivalent Expressions 6