Question P.6.4: Dividing Rational Expressions Divide: x² - 2x - 8/x² - 9 ÷ x...
Dividing Rational Expressions
Divide: \frac{x² – 2x – 8}{x² – 9} \div \frac{x – 4}{x + 3}.
This is the given division problem.
\begin{array}{ll}\frac{x² – 2x – 8}{x² – 9} \cdot \frac{x+3}{x -4}&\text{Invert the divisor and multiply.}\\ \frac{(x-4)(x+2)}{(x+3)(x-3)} \cdot \frac{x+3}{x -4}&\text{Factor as many numerators and denominators}\\&\text{ as possible. For nonzero denominators, }x ≠ -3,x ≠ 3,\\&\text{ and }x ≠ 4.\\\frac{\overset{1}{\cancel{(x-4)}}(x+2)}{\underset{1}{\bcancel{(x+3)}}(x-3)} \cdot \frac{\overset{1}{\bcancel{(x+3)}}}{\underset{1}{\cancel{(x-4)}}}&\text{Divide numerators and denominators by common factors.}\\\frac{(x+2)}{(x-3)}, x ≠ -3, x ≠ 3, \text{and }x ≠ 4.&\text{ Multiply the remaining factors in}\\ &\text{the numerators and in the denominators.}\end{array}
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