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Algebra II

Multiplying and Dividing Rational Expressions

Lesson

Multiplying and dividing rational expressions follows the same rules as numeric fractions, but with polynomials.

Multiplying

AB⋅CD=A⋅CB⋅D\frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D}

Best practice: factor everything first, cancel any common factors across the two fractions, then multiply what’s left.

Dividing

AB÷CD=AB⋅DC\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}

Flip the second fraction (take its reciprocal), then multiply.

Worked example 1 — multiplying

x2−4x+3⋅x+3x−2\frac{x^2 - 4}{x + 3} \cdot \frac{x + 3}{x - 2}

Factor numerators:

(x+2)(x−2)x+3⋅x+3x−2\frac{(x+2)(x-2)}{x+3} \cdot \frac{x+3}{x-2}

Cancel (x-2) and (x+3):

=x+2= x + 2

Worked example 2 — dividing

x2−9x+1÷x−3x2−1\frac{x^2 - 9}{x + 1} \div \frac{x - 3}{x^2 - 1}

Flip the second fraction:

=x2−9x+1⋅x2−1x−3= \frac{x^2 - 9}{x + 1} \cdot \frac{x^2 - 1}{x - 3}

Factor and cancel:

=(x+3)(x−3)x+1⋅(x+1)(x−1)x−3= \frac{(x+3)(x-3)}{x+1} \cdot \frac{(x+1)(x-1)}{x-3}
=(x+3)(x−1)= (x + 3)(x - 1)

How to type your answer

If the result simplifies to a polynomial, write the expanded form (no parens). If it’s a fraction, use / with parens. Examples: x+2, x^2+2x-3, (x-1)/(x+4).

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

x2⋅4x\frac{x}{2} \cdot \frac{4}{x}

Problem 2

x+13⋅6x+1\frac{x+1}{3} \cdot \frac{6}{x+1}

Problem 3

x2x+1÷xx+1\frac{x^2}{x+1} \div \frac{x}{x+1}

Problem 4

x−25⋅10x−2\frac{x-2}{5} \cdot \frac{10}{x-2}

Practice

Standard problems matching the lesson.

Problem 5

x2−4x+3⋅x+3x−2\frac{x^2 - 4}{x + 3} \cdot \frac{x + 3}{x - 2}

Problem 6

x2−9x+2⋅x+2x+3\frac{x^2 - 9}{x + 2} \cdot \frac{x + 2}{x + 3}

Problem 7

x2+5x+6x−4⋅x−4x+2\frac{x^2 + 5x + 6}{x - 4} \cdot \frac{x - 4}{x + 2}

Problem 8

x2−25x+1⋅x+1x+5\frac{x^2 - 25}{x + 1} \cdot \frac{x + 1}{x + 5}

Problem 9

x+2x−1÷x+2x+3\frac{x + 2}{x - 1} \div \frac{x + 2}{x + 3}

Problem 10

x2−1x+4÷x−1x+4\frac{x^2 - 1}{x + 4} \div \frac{x - 1}{x + 4}

Problem 11

x2−16x+5÷x−4x+5\frac{x^2 - 16}{x + 5} \div \frac{x - 4}{x + 5}

Problem 12

x2−9x+1÷x−3x2−1\frac{x^2 - 9}{x + 1} \div \frac{x - 3}{x^2 - 1}

Problem 13

x+3x−5⋅x2−25x2−9\frac{x + 3}{x - 5} \cdot \frac{x^2 - 25}{x^2 - 9}

Problem 14

2xx+1⋅x+1x\frac{2x}{x + 1} \cdot \frac{x + 1}{x}

Problem 15

x2+4x+4x−1⋅x−1x+2\frac{x^2 + 4x + 4}{x - 1} \cdot \frac{x - 1}{x + 2}

Problem 16

6x2−4⋅x+22\frac{6}{x^2 - 4} \cdot \frac{x + 2}{2}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

x2+2x−3x2+5x+6⋅x+2x−1\frac{x^2 + 2x - 3}{x^2 + 5x + 6} \cdot \frac{x + 2}{x - 1}

Problem 18

x2−4x2+4x+4÷x−2x+2\frac{x^2 - 4}{x^2 + 4x + 4} \div \frac{x - 2}{x + 2}

Problem 19

2x+6x−1⋅x2−1x+3\frac{2x + 6}{x - 1} \cdot \frac{x^2 - 1}{x + 3}

Problem 20

x2−9x2−4⋅x+2x+3\frac{x^2 - 9}{x^2 - 4} \cdot \frac{x + 2}{x + 3}

Problem 21

x2+7x+10x2−1÷x+5x−1\frac{x^2 + 7x + 10}{x^2 - 1} \div \frac{x + 5}{x - 1}

Problem 22

4x2−1x2−4⋅x+22x+1\frac{4x^2 - 1}{x^2 - 4} \cdot \frac{x + 2}{2x + 1}

Practice

Standard problems matching the lesson.

Problem 23

Rectangle: width (x² − 9)/(x + 3), length (x + 3)/(x − 3). Area, simplified.

Problem 24

Simplify (x + 1)/2 · 6/(x + 1).

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Simplify (x² − 4)/(x² − 1) · (x + 1)/(x + 2).

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