Problem 144 - College Algebra Essentials

Question

Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$\frac{x^{2}+6 x+5}{x^{2}-25}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{x+1}{x-5}\).

Step 1: Factorize the Numerator

The numerator is a quadratic polynomial which can be factorized. Let's write the polynomial \((x^{2}+6x+5)\) in factorized form by deciding two numbers that multiply to give the constant term (+5) and that add up to give the coefficient of the middle term (+6). The numbers that satisfy this are +5 and +1. Therefore, the numerator \((x^{2}+6x+5)\) can be factorized as \((x+5)(x+1)\).

Step 2: Factorize the Denominator

The denominator is a difference of squares which is another standard form of a second degree polynomial that can be easily factorized. We can write \((x^{2}-25)\) as \((x+5)(x-5)\). Here, the numbers +5 and -5 multiply to give -25 and subtract to give 0 (which matches since there is no 'x' term).

Step 3: Division of Common Factors and Simplification

Now that both the numerator and the denominator are factorized, we can divide out the common factor from the numerator and the denominator. Here, the common factor is \((x+5)\). When we divide out this factor from both the numerator and the denominator, we get a simplified form of the original expression as \(\frac{x+1}{x-5}\).