Simplify, multiply, divide, add, and subtract algebraic fractions, and solve equations that contain them — the same fraction rules from arithmetic, with polynomials underneath. Every question comes with a written explanation of the factoring step that made it possible.
(x + 3)/(x + 5) can't have the x's canceled — only a factor that multiplies the entire numerator and denominator can be canceled, not a term buried inside a sum.
An expression can simplify to something that looks defined everywhere, but the original denominator's zeros are still excluded from the domain.
1/(x + 1) + 1/(x − 1) isn't 2/(x² − 1) — each numerator has to be scaled by the other denominator first, exactly like adding 1/2 + 1/3 by hand.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
Simplify: (x² − 9)/(x + 3)
A — x − 3. Factor the numerator: x² − 9 = (x − 3)(x + 3), a difference of squares. The (x + 3) factor cancels with the denominator, leaving x − 3 — but only for x ≠ −3, since that value made the original denominator zero and is excluded from the domain even though it no longer appears in the simplified form. x + 3 (B) comes from canceling the wrong factor.
Multiply and simplify: (x² − 4)/(x + 5) · (x + 5)/(x − 2)
A — x + 2. Factor first: x² − 4 = (x − 2)(x + 2). The (x + 5) factors cancel across the product, and the (x − 2) in the second denominator cancels with the (x − 2) from the factored numerator, leaving x + 2 — valid for x ≠ −5 and x ≠ 2, the values that zero out the original denominators. (x + 2)(x + 5) (C) comes from canceling only the (x − 2) factors and forgetting the (x + 5) also cancels across the two fractions.
Solve for x: 1/(x − 2) + 1/(x + 2) = 4/(x² − 4)
A — No solution. Multiplying every term by the least common denominator (x − 2)(x + 2) clears the fractions: (x + 2) + (x − 2) = 4, which simplifies to 2x = 4, so x = 2. But x = 2 makes the original denominators x − 2 and x² − 4 equal to zero — it's excluded from the domain before you even solve, so it can't be a valid solution. The equation has no solution. x = 2 (B) is what you get by stopping at the algebra and skipping the domain check that rational equations always require.
Rational expressions combine two skills — factoring and fraction rules — so a mistake here is almost always one of those two, not a new one to learn.
Factor numerator and denominator completely as the very first move, before simplifying, multiplying, or adding. Untimed practice is where that ordering becomes automatic.
Move to timed sessions once factoring-first is a reflex. Rational equations run longest in this topic since there's a domain check at the end.
Pair rational expressions with polynomials in a mock — factoring is the shared skill underneath both.
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