Dividing Rational Functions

Dividing rational functions uses the same rule as dividing any fractions: invert the second fraction and multiply. After that, apply the same factor-and-cancel approach used in multiplication.

The Method

  1. Invert the second fraction (the divisor).
  2. Change the division to multiplication.
  3. Factor every numerator and denominator.
  4. Cancel common factors.
  5. Multiply what remains.

As always, track every value of \(x\) that made any denominator zero — those remain excluded from the domain.

Examples

Example 1

$$\frac{x+1}{x+3} \div \frac{3x+3}{x-2}$$

Invert and multiply:

$$\frac{x+1}{x+3} \cdot \frac{x-2}{3x+3}$$

Factor: \(3x + 3 = 3(x+1)\). Rewrite:

$$\frac{(x+1)}{(x+3)} \cdot \frac{(x-2)}{3(x+1)}$$

Cancel \((x+1)\):

$$\frac{\cancel{(x+1)}}{(x+3)} \cdot \frac{(x-2)}{3\cancel{(x+1)}} = \frac{x-2}{3(x+3)}$$

Domain: \(x \neq -3, -1, 2\). (\(x = 2\) was excluded by the original divisor's denominator before inverting.)

Example 2

$$\frac{x^2 - 9}{x^2 + 5x + 4} \div \frac{x - 3}{x + 1}$$

Invert and multiply:

$$\frac{x^2 - 9}{x^2 + 5x + 4} \cdot \frac{x+1}{x-3}$$

Factor:

  • \(x^2 - 9 = (x-3)(x+3)\)
  • \(x^2 + 5x + 4 = (x+1)(x+4)\)

$$\frac{(x-3)(x+3)}{(x+1)(x+4)} \cdot \frac{(x+1)}{(x-3)}$$

Cancel \((x-3)\) and \((x+1)\):

$$\frac{\cancel{(x-3)}(x+3)}{\cancel{(x+1)}(x+4)} \cdot \frac{\cancel{(x+1)}}{\cancel{(x-3)}} = \frac{x+3}{x+4}$$

Domain: \(x \neq 3, -1, -4\).

Example 3

$$\frac{4x^2 - 1}{2x^2 + x} \div \frac{2x - 1}{x}$$

Invert and multiply:

$$\frac{4x^2 - 1}{2x^2 + x} \cdot \frac{x}{2x-1}$$

Factor:

  • \(4x^2 - 1 = (2x-1)(2x+1)\)
  • \(2x^2 + x = x(2x+1)\)

$$\frac{(2x-1)(2x+1)}{x(2x+1)} \cdot \frac{x}{2x-1}$$

Cancel \((2x-1)\), \((2x+1)\), and \(x\):

$$= 1 \qquad x \neq 0,\ \frac{1}{2},\ -\frac{1}{2}$$

Practice Problems

1. Divide: \(\dfrac{x^2-1}{x+2} \div \dfrac{x-1}{x+2}\) Show answerInvert: \(\frac{x^2-1}{x+2} \cdot \frac{x+2}{x-1}\). Factor \(x^2-1 = (x-1)(x+1)\). Cancel \((x-1)\) and \((x+2)\): result is \(x+1\), with \(x \neq 1, -2\).

2. Divide: \(\dfrac{x^2+x-6}{x^2-4} \div \dfrac{x+3}{x+2}\) Show answerInvert: multiply by \(\frac{x+2}{x+3}\). Factor: numerator \((x+3)(x-2)\), denominator \((x-2)(x+2)\). After canceling \((x+3)\), \((x-2)\), and \((x+2)\): result is \(1\), with \(x \neq 2, -2, -3\).

3. Divide: \(\dfrac{3x^2-12}{x^2+x} \div \dfrac{x-2}{x}\) Show answerInvert: multiply by \(\frac{x}{x-2}\). Factor: \(3x^2-12 = 3(x-2)(x+2)\), \(x^2+x = x(x+1)\). Cancel \((x-2)\) and \(x\): result is \(\frac{3(x+2)}{x+1}\), with \(x \neq 0, 2, -1\).

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