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Equivalent Expressions Flashcards

6 cards from real Bluebook SAT Test practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

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

    Answer: (2x − 1) / (x + 1)

    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).

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

    Answer: (2x − 5)² − 6

    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.

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

    Answer: (y − x) / (y + x)

    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.

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

    Answer: (x² + 2x + 4) / (x + 2)

    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).

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

    Answer: 9x⁴y²

    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.

  6. When (2x³ − 3x² + x − 5) is divided by (x − 2), the result can be written in the form Q(x) + R/(x − 2). Which of the following gives the correct quotient Q(x) and remainder R?

    Answer: Q(x) = 2x² + x + 3, R = 1

    Using polynomial long division: 2x³ ÷ x = 2x²; multiply back and subtract to get x² + x − 5. Then x² ÷ x = x; multiply and subtract to get 3x − 5. Then 3x ÷ x = 3; multiply and subtract to get remainder 1. So Q(x) = 2x² + x + 3 and R = 1. Verify: (2x² + x + 3)(x − 2) + 1 = 2x³ − 3x² + x − 5 ✓.