Grade by Grade
A. Characteristics of Functions · Functions and Equivalent Expressions

Multiplying and Dividing Rational Expressions

A3.3About 70 minutes
Learning goals
  • I can multiply rational expressions by factoring and dividing out common factors.
  • I can divide rational expressions by multiplying by the reciprocal.
  • I can state all restrictions, including those from the divisor's numerator.
  • I can use rational expressions in simple rate and area problems.

Words to know

reciprocal
The fraction flipped: the reciprocal of A/B is B/A (A, B ≠ 0).
product / quotient
The result of multiplying / dividing.
divisor
The expression you divide by.
restriction
A value that makes any denominator zero at any step.

Same rules as fractions

To multiply fractions you multiply tops and bottoms; to divide, you multiply by the reciprocal ("keep, change, flip"). And the smart move is to cancel before you multiply, so the numbers stay small:

(4/9) × (15/8) = (4 × 15)/(9 × 8): cancel 4 with 8 (leaves 1 and 2) and 15 with 9 (leaves 5 and 3), giving 5/6.

Rational expressions work exactly the same way. Because the "numbers" are now polynomials, "cancel first" means factor first.

1. Factoreverything2. Restrictall denominators3. Flip(dividing only)4. Cancelcommon factors5. Multiplywhat is leftDividing: also restrict the NUMERATOR of the divisor,because after the flip it becomes a denominator.
The five steps for multiplying or dividing rational expressions.
Worked example

Example 1: Monomials

Simplify (6x²/(5y)) × (10y³/(3x)) and state the restrictions.

  1. Restrictions: y ≠ 0 (first denominator) and x ≠ 0 (second denominator).
  2. Multiply: 60x²y³ / (15xy).
  3. Simplify: 60/15 = 4, x²/x = x, y³/y = y². Result: 4xy², x ≠ 0, y ≠ 0.

4xy², x ≠ 0, y ≠ 0

Worked example

Example 2: Multiplying with factoring

Simplify (x² − 4)/(x² + 5x + 6) × (x + 3)/(x − 2).

  1. Factor: (x − 2)(x + 2) / [(x + 2)(x + 3)] × (x + 3)/(x − 2).
  2. Restrictions from the denominators: x ≠ −2, −3, 2.
  3. Cancel (x − 2), (x + 2) and (x + 3): everything cancels and 1 is left.
  4. So the product equals 1 for every allowed x.

1, x ≠ −3, −2, 2

Dividing

To divide by a rational expression, multiply by its reciprocal:

(A/B) ÷ (C/D) = (A/B) × (D/C)

There is an extra restriction. Before the flip, B and D cannot be zero. After the flip, C is in a denominator, so C cannot be zero either. Find all restrictions before you cancel.

Worked example

Example 3: Dividing

Simplify (x² − 9)/(x² − x − 6) ÷ (x² + 4x + 3)/(x² + x − 2).

  1. Factor everything: (x − 3)(x + 3)/[(x − 3)(x + 2)] ÷ (x + 1)(x + 3)/[(x + 2)(x − 1)].
  2. Restrictions: first denominator x ≠ 3, −2; divisor's denominator x ≠ −2, 1; divisor's numerator x ≠ −1, −3. Together: x ≠ 3, −2, 1, −1, −3.
  3. Flip and multiply: (x − 3)(x + 3)/[(x − 3)(x + 2)] × (x + 2)(x − 1)/[(x + 1)(x + 3)].
  4. Cancel (x − 3), (x + 3) and (x + 2): (x − 1)/(x + 1).
  5. Check with x = 2: original (−5/−4) ÷ (15/4) = 1.25 ÷ 3.75 = 1/3; answer (2 − 1)/(2 + 1) = 1/3 ✓.

(x − 1)/(x + 1), x ≠ −3, −2, −1, 1, 3

(x − 3)(x + 3)(x − 3)(x + 2)(x + 1)(x + 3)(x + 2)(x − 1)÷x ≠ −1, −3divisor numeratorx ≠ 3, −2first denominatorx ≠ −2, 1divisor denominatorAll restrictions: x ≠ 3, −2, 1, −1, −3
In a division, three parts of the expression produce restrictions. The divisor's numerator counts because it ends up in the denominator.
Worked example

Example 4: Trinomials with a ≠ 1

Simplify (2x² + 7x + 3)/(4x² − 1) ÷ (x² + 6x + 9)/(2x − 1).

  1. Factor: 2x² + 7x + 3 = (2x + 1)(x + 3); 4x² − 1 = (2x − 1)(2x + 1); x² + 6x + 9 = (x + 3)².
  2. Restrictions: 2x − 1 ≠ 0 and 2x + 1 ≠ 0 give x ≠ ±1/2; the divisor's numerator gives x ≠ −3.
  3. Flip: (2x + 1)(x + 3)/[(2x − 1)(2x + 1)] × (2x − 1)/(x + 3)².
  4. Cancel (2x + 1), (2x − 1) and one (x + 3): 1/(x + 3).

1/(x + 3), x ≠ −3, ±1/2

Worked example

Example 5: A rate problem

A cyclist rides (x² − 4) km at an average speed of (x + 2) km/h, for some x > 2. Write and simplify an expression for the time taken. Then find the time when x = 7.

  1. time = distance ÷ speed = (x² − 4)/(x + 2).
  2. Factor and cancel: (x − 2)(x + 2)/(x + 2) = x − 2 hours (x ≠ −2, which is already excluded by x > 2).
  3. When x = 7: distance 45 km at 9 km/h, and the expression gives 7 − 2 = 5 h. Check: 45 ÷ 9 = 5 ✓.

Time = x − 2 hours; 5 h when x = 7.

Just for fun

To divide fractions, keep, change, flip. It is the only time in math class when doing a flip is encouraged.

Hands-on activity

Fraction-strip check

You need: paper, a ruler, scissors (take care, or ask a grown-up) and a calculator.

  1. Cut a strip 24 cm long. Mark it in thirds (8 cm each) and shade 2/3 of it.
  2. How many 1/6 pieces (4 cm each) fit into the shaded part? Count them. This shows 2/3 ÷ 1/6.
  3. Check with keep-change-flip: (2/3) × (6/1).
  4. Now use the same rule on a rational expression: simplify (x² − 1)/x ÷ (x + 1)/x², then test your answer with x = 3 on a calculator.

What to look for: the shaded part (16 cm) holds four 4 cm pieces, and (2/3) × 6 = 4 ✓. For the expression: (x − 1)(x + 1)/x × x²/(x + 1) = x(x − 1), with x ≠ 0, −1. With x = 3: (8/3) ÷ (4/9) = (8/3) × (9/4) = 6, and 3(2) = 6 ✓.

Your turn

1

Simplify (8a³/(3b)) × (9b²/(4a)).

2

Simplify (x + 5)/(x − 1) × (x² − 1)/(x² + 5x).

3

Simplify (12x²/(5y)) ÷ (4x/(15y²)).

4

Simplify (x² − 16)/(3x) ÷ (x + 4)/(6x²).

5

State all restrictions for (x + 2)/(x − 3) ÷ (x − 5)/(x + 1).

6

Simplify (x² + 3x + 2)/(x² − 9) × (x − 3)/(x + 1).

More practice: warm-up

1

Compute (3/4) × (8/9).

2

Compute (5/6) ÷ (10/3).

3

What is the reciprocal of (x − 2)/(x + 7)?

4

Simplify (x/y) × (y/x).

5

Simplify 2x ÷ (x/5).

More practice: on your own

1

Simplify (x² − 2x − 15)/(x² − 9) × (x² + x − 6)/(x² − 7x + 10).

2

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

3

Simplify (x² + 6x + 9)/(x² − 4x) ÷ (x² − 9)/(x² − 16).

4

Simplify (5 − x)/(x + 2) × (x² − 4)/(x² − 25).

5

A rectangle has area (x² + 7x + 10) m² and length (x + 5) m. A second rectangle has the same width and length (x + 1) m. Write and simplify the second rectangle's area.

Challenge

1

Simplify (x³ − x)/(x² + 2x + 1) × (x + 1)/(x² − x).

2

Simplify [(x² − y²)/(xy)] ÷ [(x − y)/y²].

3

Simplify (a² − 5a + 6)/(a² − 4) ÷ (a² − 6a + 9)/(a² + 2a) × (a − 3)/a.

Check your understanding

Quiz

1.

Simplify (x² − 1)/x × x²/(x − 1).

2.

To divide by (x + 3)/(x − 4), you multiply by:

3.

Which restriction comes ONLY from dividing in (x/(x − 2)) ÷ ((x + 5)/(x + 1))?

4.

Simplify (4x³/(9y²)) ÷ (2x/(3y)).

5.

Simplify (x² + 5x + 6)/(x + 3) × 1/(x + 2).

6.

Which step should come FIRST when multiplying rational expressions?

For parents and teachers

Big idea: Expectation A3.3 includes multiplying and dividing rational expressions and stating the restrictions on the variable.

How to help: Have your teen say the steps aloud: factor, restrict, flip (if dividing), cancel, multiply. Checking with a test value such as x = 2 catches most mistakes.

Common difficulties:

  • Multiplying everything out before cancelling (huge, messy polynomials).
  • Forgetting the divisor's numerator when listing restrictions.
  • Flipping the wrong fraction (flip only the one you are dividing by).
  • Cancelling terms rather than factors.