SAT Equivalent Expressions Questions
Rewriting an expression without changing its value.
In short
- Equivalent Expressions is part of Advanced Math, about 35% of the scored Math questions: 13 to 15 on every test.
- Math questions from all four areas are mixed in each module and arranged from easiest to hardest.
- Polynomial identity is the trap Rivie's practice tests set most often in Equivalent Expressions questions.
Common traps
Most common first, from Rivie's ten practice tests.
Polynomial identity
Match coefficients across an identity (a polynomial that holds for all x); the wrong choice solves only one coefficient equation and stops.
The usual mistake: You expand the left side, match the x² coefficient correctly, and pick that, missing that the constant term gives a different (correct) answer.
Factor signs
Quadratic factoring where the constants pair correctly but you assigned the wrong signs (+/−) to the factors.
The usual mistake: You factor x² − 5x + 6 as (x+2)(x+3); you found the right magnitudes but not the signs that produce the correct middle term.
Exponent multiply vs. divide
x^a · x^b = x^(a+b), but x^a / x^b = x^(a−b). The wrong choice multiplies the exponents in one case and divides in the other inconsistently.
The usual mistake: You confuse the addition rule with the multiplication rule (which is for (x^a)^b = x^(ab)).
Rational factor cancel
Simplify (x² − 4)/(x − 2) by factoring and cancelling; the wrong choice cancels terms instead of factors (e.g. cancels the 2's in numerator and denominator).
The usual mistake: You cancel matching tokens you see, not equivalent expressions you've factored.
Exponent rules: multiply vs. add
Sibling of exponent_multiply_vs_divide: confusing (x^a)^b = x^(ab) with x^a · x^b = x^(a+b).
The usual mistake: You see two exponents stacked and pick whichever rule you remembered last.
FOIL cross terms
Expanding (a + b)(c + d): the wrong choice gets the outside terms ac and bd but forgets the inside cross terms ad and bc.
The usual mistake: You distribute first × first and last × last and call it done; the FOIL middle gets dropped.
Distribute and combine
When simplifying expressions like 3(x + 2) − (x − 5), distribute both factors first, THEN combine like terms. Skipping a step yields a sign or coefficient mistake.
The usual mistake: You combine before distributing, or distribute only the first factor and forget the second. The trap punishes shortcuts on multi-step simplifications.
Distribute negative
When a minus sign precedes parentheses (e.g., 5 − (x − 3)), the minus distributes to EVERY term inside, flipping each sign: 5 − x + 3, not 5 − x − 3.
The usual mistake: You only flip the first term inside the parens and leave the rest unchanged. Sign errors here cascade into wrong final answers across the whole problem.
Ignored constant term
When manipulating an equation (factoring, completing the square, clearing fractions), the constant term gets dropped or treated as zero.
The usual mistake: You focus on the variable terms and forget to apply the same operation to the constant. The fix is to keep both sides of the equation visually balanced after every step.
Try a question
From Rivie's practice tests. Pick an answer to see why it is right or wrong.
Which of the following is equivalent to x² + 7x + 10?
Video lessons
Free on YouTube, picked for this topic.