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Division of Rational Expressions

Evaluate 3 x2...

Question

Evaluate $\frac{3 x^{2} - x - 4}{9 x^{2} - 16} \div \frac{4 x^{2} - 4}{3 x^{2} - 2 x - 1}$

A

\frac{x + 1}{4 (x + 4)}

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B

\frac{3 x + 1}{4 (3 x + 4)}

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C

\frac{3 x + 5}{4 (3 x + 5)}

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D

None of the above

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Solution

The correct option is B $\frac{3 x + 1}{4 (3 x + 4)}$
$\frac{3 x^{2} - x - 4}{9 x^{2} - 16} \div \frac{4 x^{2} - 4}{3 x^{2} - 2 x - 1}$

= $\frac{3 x^{2} - x - 4}{9 x^{2} - 16} \times \frac{3 x^{2} - 2 x - 1}{4 x^{2} - 4}$

= $\frac{3 x^{2} + 3 x - 4 x - 4}{(3 x)^{2} - (4)^{2}} \times \frac{3 x^{2} - 3 x + x - 1}{4 (x^{2} - 1)}$

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

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

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

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