Question

Using complete sentences, explain each step in simplifying the ratio x^2+8x/x^2+10x+16 and provide the simplified form.

Answer 1

To simplify the rational expression [x^2+8x]/[x^2+10x+16], we take the numerator and show it as a prduct of two factors. Then we take denominator and show it as product of two factors. We examine for any common factor between numerator and denominator. Then we divide (or reduce or cancel) both numerator and denominator by the common factor. That completes the simplification of the rational expression.

Numerator, x^2 +8x = x(x+8).

Denominator , x^2+10x+16 = x^2+8x+2x+16 .

x^2+10x+16= x(x+8)+2(x+8).

x^2+10x+16 = (x+8)(x+2).

Therefore, the rational expression, (x^2+8x)/(x^2+10x+16) = x(x+8)/{(x+8)(x+2)}. Now we find that x+8 is a common factor of both numerator and denominator of the rational expression on left. So we divide both numerator and denominator by x+8 and rewrite the given rational expression and its equivalent rational expression in the simplified form:

(x^2+8x)/(x^2+10x+16) = x/(x+2).