Lesson · Rational and Radical Expressions · Rational expressions

Multiplying and Dividing Rational Expressions

Multiplication of algebraic fractions is exactly multiplication of numerical ones, and division is multiplication by the reciprocal. The only genuinely new thing is that dividing forbids more values than it looks like it forbids.

Open the interactive lesson Practice

a/b · c/d = (ac)/(bd) a/b ÷ c/d = a/b · d/c the division excludes zeros of b, of d, AND of c factor everything, cancel across, multiply last

Two rules, three sources of exclusions

Core ideas

What this lesson establishes

No common denominator is needed here — that is “Adding and Subtracting Rational Expressions”'s problem. What this lesson demands is that you factor first and that you account for a divisor which may itself be zero.

Multiplication is across the top and across the bottom

a/b · c/d = (ac)/(bd), with no adjustment of denominators. It is the one fraction operation that needs no preparation, which is why division is converted into it.

Division is multiplication by the reciprocal

a/b ÷ c/d = a/b · d/c. Dividing by c/d asks what you must multiply c/d by to reach a/b, and d/c undoes c/d because their product is 1.

Cancel before you multiply, not after

The factors are already visible before multiplying and buried afterwards. Expanding (x − 2)(x + 2)(x + 3) first turns a one-line cancellation into a cubic you then have to factor again.

The material

Products, quotients, and the exclusion that hides

Two theorems, then the value that only division forbids.

Theorem · Product rule

For rational expressions with b ≠ 0 and d ≠ 0, (a/b)·(c/d) = (ac)/(bd). The domain of the product excludes every zero of b and every zero of d.

Theorem · Quotient rule

For b ≠ 0, d ≠ 0 and c ≠ 0, (a/b) ÷ (c/d) = (a/b)·(d/c) = (ad)/(bc). The three conditions are each doing work, and the third is the one that is easy to lose.

Why c ≠ 0? Because a/b ÷ c/d means dividing by the number c/d, and that number is zero exactly when c is zero. Dividing by zero is undefined whether the zero arrives as a plain 0 or as a fraction with a zero on top.

Example · The hidden exclusion

In (x² − 4)/(x² + 6x + 9) ÷ (x − 2)/(x + 3), the visible denominators (x + 3)² and x + 3 forbid x = −3. But the divisor (x − 2)/(x + 3) is zero at x = 2, so x = 2 is forbidden too — and after you flip, x − 2 sits in a denominator where you can finally see it.

This is the general pattern: flipping the divisor turns its numerator into a denominator, and a denominator is something you already know how to read. Record the exclusion at the moment of flipping, because the factor x − 2 is about to cancel and disappear.

With the exclusions safely written down, the mechanical part is short. Factor every numerator and denominator, cancel any factor that appears on both levels of the combined fraction, and only then write the product.

Example · A product

(x + 1)/(x − 5) · (x − 5)/(x + 2). The factor x − 5 is on top of the second and on the bottom of the first, so it cancels: the answer is (x + 1)/(x + 2) with x ≠ 5 and x ≠ −2. Multiplying first would have produced (x² − 4x − 5)/(x² − 3x − 10) and the same cancellation, harder.

There is no rule about numerators here to match the one about denominators in “Adding and Subtracting Rational Expressions”. A product needs no common denominator, so the temptation to find one is worth resisting: it is extra work that cannot help.

Interactive

Multiplying and dividing

Invert and multiply, and the exclusion that move creates

Cancel across the product

Enter two expressions and choose multiply or divide. The lab flips the divisor, lists every exclusion including the one the divisor contributes, and cancels factor by factor.

Worked example

First expression

Operation

Second expression

——

Type any rational expression in x: 2x, (x+1)(x-2) and 1/x + 1/2 all parse. Every coefficient below is an exact fraction, and every excluded value is found from the denominators as you wrote them.

Method

Multiplying or dividing

For division, step one is the flip; everything after is the same.

If it is a division, flip the divisor now

Rewrite ÷ c/d as · d/c before doing anything else, so that the rest of the problem is a single product.

Factor all four polynomials

Every numerator and every denominator, completely. Nothing can be cancelled until this is done.

List the exclusions, divisor included

Zeros of both original denominators, plus the zeros of the divisor's numerator if you divided. That last group vanishes from the answer.

Cancel, then multiply what is left

Strike common factors across the whole product, then write the remaining factors on each level. Leave the answer factored.

Across, or flipped

Score 0 / 0

Choose the answer you can justify.

Divide (x² − 4)/(x² + 6x + 9) by (x − 2)/(x + 3)

A division with three separate exclusions, one of which is invisible in the final answer.

(x² − 4)/(x² + 6x + 9) ÷ (x − 2)/(x + 3) flip the divisor: = (x² − 4)/(x² + 6x + 9) · (x + 3)/(x − 2) factor: x² − 4 = (x − 2)(x + 2) x² + 6x + 9 = (x + 3)² exclusions: x ≠ −3 (denominators) x ≠ 2 (divisor) = ((x − 2)(x + 2)(x + 3)) / ((x + 3)²(x − 2)) = (x + 2)/(x + 3), x ≠ −3, x ≠ 2 check at x = 0 : (−4/9) ÷ (−2/3) = (−4/9)(−3/2) = 2/3 (x + 2)/(x + 3) = 2/3 ✓

Both x − 2 and one x + 3 cancelled, and with them went every trace of why 2 is not allowed. (x + 2)/(x + 3) evaluated at x = 2 gives 4/5, a number the original division cannot produce, because at x = 2 it asks you to divide by 0.

For a faded quotient, simplify (x² − 1)/(x² + 3x + 2) ÷ (x − 1)/(x + 2). The supplied first move is to flip the divisor and immediately add x ≠ 1 to the exclusions. Factor, cancel and record the two original denominator zeros yourself. The result is 1, with x ≠ −2, −1, 1; missing 1 means the divisor's numerator was never checked, while missing −1 means a cancelled original factor erased its restriction.

Common mistakes

Forgetting that the divisor may be zeroIn a/b ÷ c/d the zeros of c are excluded, and they usually cancel out of sight during the simplification. Write them down when you flip.

Looking for a common denominatorProducts and quotients never need one. Building it wastes time and, worse, invites you to add the numerators out of habit.

Multiplying everything out before cancellingExpanding first is legal and almost always a mistake: it turns visible factors into a polynomial you must factor again to finish.

Check your understanding

✓

Finish when division is automatic and its extra exclusion is too.

You should convert a division to a product without thinking, and still stop to record the zeros of the divisor's numerator. The arithmetic here is the easiest in the course; the bookkeeping is what distinguishes a correct answer from a nearly correct one.

Note: The lab shows the flip as its own step rather than folding it into the cancellation, and it labels each exclusion with the factor that caused it. When a factor cancels, its exclusion stays on the list in a dimmed row — the answer changes, the domain does not.