Equivalent Expressions Flashcards

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

Answer: (x² + 2x + 4)/(x + 2)

Factor the numerator as a 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 B incorrectly uses the difference-of-cubes pattern for both parts. Choice C over-cancels. Choice D uses the wrong denominator.

Which expression is equivalent to 3x² − 12x + 7?

Answer: 3(x − 2)² − 5

Complete the square: factor 3 from the first two terms → 3(x² − 4x) + 7. Complete inside the parentheses: x² − 4x = (x − 2)² − 4. Substitute back: 3[(x − 2)² − 4] + 7 = 3(x − 2)² − 12 + 7 = 3(x − 2)² − 5. Choice B forgets to subtract 3·4 = 12. Choice C uses the wrong vertex. Choice D incorrectly distributes the leading coefficient.

Which expression is equivalent to (x^(1/3) + y^(1/3))(x^(2/3) − x^(1/3)·y^(1/3) + y^(2/3))?

Answer: x + y

Let a = x^(1/3) and b = y^(1/3). The expression becomes (a + b)(a² − ab + b²), which is the sum-of-cubes factoring pattern in reverse: a³ + b³. Substituting back: (x^(1/3))³ + (y^(1/3))³ = x + y. The other choices confuse the pattern with difference of squares or partial factorizations.

When (2x³ + 3x² − 5x + 1) is divided by (x + 2), which expression is equivalent to the result?

Answer: 2x² − x − 3 + 7/(x + 2)

Using synthetic division with root −2 on coefficients [2, 3, −5, 1]: bring down 2; 2·(−2) = −4, 3+(−4) = −1; (−1)·(−2) = 2, −5+2 = −3; (−3)·(−2) = 6, 1+6 = 7. The quotient is 2x² − x − 3 with remainder 7, giving 2x² − x − 3 + 7/(x+2). Choice C uses a negative remainder. Choices B and D result from arithmetic errors in the synthetic division steps.

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

Answer: −1/[x(x + h)]

Combine the fractions in the numerator: [x − (x + h)] / [x(x + h)] = −h / [x(x + h)]. Dividing by h gives −1/[x(x + h)]. Choice B drops the negative sign. Choice C is the limit as h → 0, not the expression itself. Choice D fails to divide by h (it multiplies instead).

For x > 1, which expression is equivalent to √(x + 2√(x − 1))?

Answer: √(x − 1) + 1

Rewrite the radicand by splitting x as (x − 1) + 1: x + 2√(x − 1) = (x − 1) + 2√(x − 1)·1 + 1 = (√(x − 1) + 1)². Taking the square root (positive, since x > 1): √(x + 2√(x − 1)) = √(x − 1) + 1. Choice B incorrectly keeps x unsplit under a square root. Choices C and D produce negative or incorrect values outside the valid domain.