Problem 52 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{x^{2}-4}{x^{2}+3 x-10} \div \frac{x^{2}+5 x+6}{x^{2}+8 x+15}$$

Step-by-Step Solution

Answer

The result of the division is 1.

Step 1: Factorize the polynomials

Rewrite the polynomials in factorized form: a) \(x^{2}-4 = (x-2)(x+2)\), b) \(x^{2}+3x-10 = (x-2)(x+5)\), c) \(x^{2}+5x+6 = (x+2)(x+3)\), d) \(x^{2}+8x+15 = (x+3)(x+5)\).

Step 2: Rewrite the division operation

A division operation is the same as multiplying by the reciprocal, so the expression can be rewritten as \((x-2)(x+2) / (x-2)(x+5) * (x+3)(x+5) / (x+2)(x+3)\).

Step 3: Simplify the expression

We can cancel out common terms that appear in both numerator and denominator. Cancel out \(x-2\) in the first fraction and \(x+2\) in the second. Also, cancel out \(x+5\) in the second fraction and \(x+3\) in tje first fraction, which leaves 1 as the final result.