Equivalent Expressions
Help Questions
SAT Math › Equivalent Expressions
Which of the following expressions is equivalent to $2x^3 - 5x^2 - 2x + 5$?
$(2x - 5)(x - 1)(x + 1)$
$(2x + 5)(x - 1)(x + 1)$
$(x - 5)(2x^2 - 1)$
$(2x - 5)(x^2 + 1)$
Explanation
Factor by grouping: $(2x^3 - 5x^2) - (2x - 5) = (2x - 5)(x^2 - 1) = (2x - 5)(x - 1)(x + 1)$. Choice A has an incorrect sign. Choices B and C are incomplete or incorrect factorizations.
Which of the following expressions is equivalent to $\frac{x^2 - 9}{x - 3}$ for $x \ne 3$?
$x + 3$
$x - 3$
$x^2 - 3$
$x + 9$
Explanation
Factor the numerator as $(x - 3)(x + 3)$ and cancel $x - 3$ to get $x + 3$. Choice A mistakes cancellation as subtraction. Choices B and C result from incorrect factoring or arithmetic.
Which of the following expressions is equivalent to $(3x - 2)(2x + 5) - (x - 4)(4x + 1)$?
$10x^2 + 9x - 14$
$2x^2 + 26x + 6$
$2x^2 - 4x - 6$
$2x^2 + 26x - 6$
Explanation
Expand each product and subtract: $(6x^2 + 11x - 10) - (4x^2 - 15x - 4) = 2x^2 + 26x - 6$. The distractors reflect sign errors or incorrect multiplication.
Which of the following expressions is equivalent to $2(3x - 5) - (x + 4) + 3$?
$7x - 11$
$5x - 11$
$5x - 1$
$5x - 17$
Explanation
Distribute and combine like terms: $6x - 10 - x - 4 + 3 = 5x - 11$. The distractors come from distributing errors or misadding constants.
Which of the following expressions is equivalent to $x^2 + 7x + 10$?
$(x+1)(x+10)$
$(x+5)(x+2)$
$(x+4)(x+3)$
$(x+5)(x-2)$
Explanation
$x^2 + 7x + 10$ factors to $(x+5)(x+2)$. The other factorizations expand to different quadratics.
Which of the following expressions is equivalent to $ (3x - 2) + (5x + 7) $?
$8x + 9$
$8x + 5$
$-2x + 5$
$8x - 5$
Explanation
Combine like terms to get $8x + 5$. The other choices reflect sign or constant addition errors.
Which of the following expressions is equivalent to $2x^2-5x-12$?
$(2x-1)(x+12)$
$(2x+3)(x-4)$
$(x+3)(2x-4)$
$(2x-3)(x+4)$
Explanation
$(2x+3)(x-4)=2x^2-8x+3x-12=2x^2-5x-12$. The other products yield incorrect middle terms or constants.
Which of the following expressions is equivalent to $\frac{6x^2y-9xy}{3xy}$?
$\frac{2x-3}{3xy}$
$2x+3$
$2x-\frac{3}{y}$
$2x-3$
Explanation
Factor $3xy$ from the numerator and cancel to get $2x-3$. A and B fail to fully cancel, and D has a sign error.
Which of the following expressions is equivalent to $(x-4)(x+5)$?
$x^2+x-20$
$x^2-x-20$
$x^2-20$
$x^2+9x-20$
Explanation
FOIL gives $x^2+5x-4x-20=x^2+x-20$. The distractors omit or miscompute the middle term.
Which of the following expressions is equivalent to $4x - 3(2x - 5) + x^2 - (x^2 - x + 1)$?
$-x-14$
$x+14$
$-3x+14$
$-x+14$
Explanation
Distribute and combine like terms: $4x-6x+15+x^2-x^2+x-1=-x+14$. The distractors reflect sign errors on the constants or mishandling the subtraction of the last parentheses.