Add, subtract, multiply, and factor polynomial expressions — factoring is where most points are won or lost, and where a single wrong sign undoes the whole answer. Every question comes with a written walkthrough of the factoring method used.
Attempting to factor a trinomial that still has a common factor in every term leads to an answer that looks wrong even when the trinomial factoring itself is done correctly.
Subtracting (3x² − 2x + 1) means flipping the sign on all three terms, not just the first one.
a² − b² factors cleanly into (a − b)(a + b), but a² + b² does not factor over the reals — the sign in the middle decides whether the pattern even applies.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
Subtract: (5x² + 3x − 4) − (2x² − x + 6)
A — 3x² + 4x − 10. Distribute the minus sign across every term of the second polynomial before combining: 5x² + 3x − 4 − 2x² + x − 6 = 3x² + 4x − 10. 3x² + 2x − 10 (B) comes from only flipping the sign on the first term of the second polynomial and forgetting to flip the sign on −x, computing 3x − x instead of 3x + x.
Factor: x² + 5x − 24
A — (x + 8)(x − 3). Find two numbers that multiply to −24 and add to 5: 8 and −3 fit both conditions, giving (x + 8)(x − 3). (x − 8)(x + 3) (B) swaps the signs on the two numbers — a factoring that would multiply to −24 correctly but add to −5, not 5, since sign placement determines which number is positive.
Factor completely: 3x³ − 12x
A — 3x(x − 2)(x + 2). First pull the GCF, 3x, from both terms: 3x(x² − 4). The remaining x² − 4 is a difference of squares and factors further into (x − 2)(x + 2), giving the fully factored form 3x(x − 2)(x + 2). 3x(x² − 4) (C) stops one step early — the GCF is correctly factored out, but the difference-of-squares pattern inside is left unfactored, which is the single most common way to lose a mark on a "factor completely" instruction.
Polynomial factoring is a pattern-recognition skill more than a formula — practice trains you to spot GCF, difference of squares, and trinomial patterns on sight.
Make pulling out the greatest common factor the first move on every problem, even when it looks unnecessary. Untimed practice is where that reflex gets built.
Move to timed sessions once GCF-checking is automatic. Factoring runs longer than simplifying since there's often more than one step.
Pair polynomials with quadratic equations in a mock — factoring a trinomial is the first step of solving most quadratics by factoring.
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