5.2.2 Multiplication and Division of Rational Expressions

5.2.2 Multiplication and Division of Rational Expressions

a Multiplication

Example 6

Simplify: $\frac{x^2 - 5x + 6}{x^2 - 25} \times \frac{x^2 + 5x}{2x - 6}$

Solution

$\frac{x^2 - 5x + 6}{x^2 - 25} \times \frac{x^2 + 5x}{2x - 6} = \frac{x^2 - 2x - 3x + 6}{x^2 - 5^2} \times \frac{x(x + 5)}{2(x - 3)}$ $= \frac{(x - 2)(x - 3)}{(x - 5)(x + 5)} \times \frac{x(x + 5)}{2(x - 3)} = \frac{x(x - 2)}{2(x - 5)}$

Example 7

Simplify: $\frac{3a^2 - 6ab}{6ab} \times \frac{10b^2}{4ab - 8b^2}$

Solution

$\frac{3a^2 - 6ab}{6ab} \times \frac{10b^2}{4ab - 8b^2}$ $= \frac{3a(a - 2b)}{6ab} \times \frac{10b^2}{4b(a - 2b)}$ $= \frac{30}{24} = \frac{5}{4}$

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Skilled Practice!
Simplify: (i) $\frac{32x^3y^2z^5}{3x^3y^3} \cdot \frac{243x^6}{x^3y^2z}$ (ii) $\frac{5a^2}{4a - 8} \cdot \frac{6a - 12}{10a}$

b Division

To divide by a rational expression, multiply by its reciprocal as in multiplication.

Example 8

$\frac{2a^2 - 2c^2}{15a^2c^2} \div \frac{3a + 3c}{5ac}$

Solution

$\frac{2a^2 - 2c^2}{15a^2c^2} \div \frac{3a + 3c}{5ac}$ $= \frac{2a^2 - 2c^2}{15a^2c^2} \times \frac{5ac}{3a + 3c}$ $= \frac{2(a^2 - c^2)}{15a^2c^2} \times \frac{5ac}{3(a + c)}$

$= \frac{2(a + c)(a - c)}{15a^2c^2} \times \frac{5ac}{3(a + c)} = \frac{2(a - c)}{9ac}$

Example 9

Simplify: $\frac{x^3 - a^3}{x^2 - 2bx + b^2} \times \frac{x^2 - bx - cx + bc}{ax - x^2} \div \frac{x^2 - cx}{x - b}$

Solution

$\frac{x^3 - a^3}{x^2 - 2bx + b^2} \times \frac{x^2 - bx - cx + bc}{ax - x^2} \div \frac{x^2 - cx}{x - b}$ $= \frac{x^3 - a^3}{x^2 - 2bx + b^2} \times \frac{x^2 - bx - cx + bc}{ax - x^2} \times \frac{x - b}{x^2 - cx}$ $= \frac{(x - a)(x^2 + ax + a^2)}{(x - b)(x - b)} \times \frac{(x - b)(x - c)}{x(a - x)} \times \frac{x - b}{x(x - c)}$ $= -(a - x)(x^2 + ax + a^2) \times \frac{1}{x(a - x)} \times \frac{1}{x}$ $= -\frac{(x^2 + ax + a^2)}{x^2}$