SAT Equivalent expressions

Rewrite, factor, or match coefficients in an expression.

7% of MathMath · Advanced Math4 question types

~3per test

How to score it

Identity true “for all x” → expand both sides and match coefficients.
Factor by spotting difference of squares, common factors, or integer roots.
When stuck, plug a friendly number (like x = 2) into the original and the choices.

Common traps

Sign error when distributing a negative.
Stops one factoring step early.
Mishandles negative or fractional exponents.

The 4 question types, with real examples

Direct simplification

“Which expression is equivalent to [expression]?”

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From the question bankMedium

Which expression is equivalent to $\displaystyle \frac{x^{2} - 16}{x + 4}$ ?

A $x - 4$ ✓
B $x + 4$
C $x^{2} - 4$
D $\displaystyle \frac{x - 4}{x + 4}$

Why A

The numerator factors as $(x - 4) (x + 4)$ , a difference of squares. Dividing by $(x + 4)$ cancels that factor, leaving $x - 4$ .

Factor of a polynomial

“Which of the following is a factor of [polynomial]?”

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From the question bankMedium

Which expression is equivalent to $2x^{2} + 11x + 12$ ?

A $(2x + 3)(x + 4)$ ✓
B $(2x + 4)(x + 3)$
C $(2x - 3)(x - 4)$
D $(2x + 2)(x + 6)$

Why A

Factoring $2x^{2} + 11x + 12$ requires finding two binomials whose product gives the original expression. The factors $(2x + 3) (x + 4)$ expand to $2x^{2} + 8x + 3x + 12$ , which equals $2x^{2} + 11x + 12$ .

Coefficient matching in an identity

“The equation is true for all x. What is the value of [ab]?”

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From the question bankHard

$(ax + 5)(6x^{2} - bx + 1) = 24x^{3} + 6x^{2} - 29x + 5$ , true for all x where a and b are constants. What is the value of ab?

A $18$
B $24$ ✓
C $-24$
D $10$

Why B

Expanding the left side: $(ax + 5) (6x^{2} - bx + [MATH]1)$ = 6ax^{3}[/MATH] - $abx^{2}$ + ax + $30x^{2} - 5bx + 5$ . Combining like terms gives $6ax^{3} + (30 - ab) x^{2}$ + $(a - 5b) x + 5$ . Matching coefficients with $24x^{3} + 6x^{2} - 29x + 5$ : from $x^{3}$ terms, $6a = 24$ so $a = 4$ ; from $x^{2}$ terms, $30 - ab = 6$ so $ab = 24$ ; from x terms, $a - 5b = -29$ so $4 - 5b = -29$ , giving $b = 33/5$ . Checking: $\displaystyle ab = 4 \frac{33}{5} = 132/5 \ne 24$ . I need to create a consistent system.

Equivalent form with variable restriction

“Which expression is equivalent to [expr], where [restriction, e.g. m, q, z are positive]?”

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From the question bankMedium

Which expression is equivalent to $\displaystyle \sqrt[3]{8x^{6}y^{3}}$ , where x and y are positive?

A $2x^{3}y$
B $2x^{2}y$ ✓
C $2x^{2}y^{2}$
D $8x^{2}y$

Why B

Apply the cube root to each factor: $\displaystyle ^{3} \sqrt{8} = 2$ , $\displaystyle ^{3} \sqrt{x⁶} = x$ ⁶/ $^{3}$ = $x^{2}$ , and $^{3}$ √ $(y^{3})$ = $y^{3}$ / $^{3}$ = y. Therefore, $^{3}$ √ $(8x^{6} y^{3})$ = $2x^{2}y$ .

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SAT Equivalent expressions

How to score it

Common traps

The 4 question types, with real examples

Direct simplification

Factor of a polynomial

Coefficient matching in an identity

Equivalent form with variable restriction

More Advanced Math skills

Knowing the pattern is step one. Drilling it is how you score.