Simplifying Rational Expressions Simplify: a. [latex]\frac{x^3+x^2}{x+1}[/latex] b. [latex]\frac{x^2+6 x+5}{x^2-25}[/latex] .

a. [latex]\frac{x^3+x^2}{x+1}=\frac{x^2(x+1)}{x+1}[/latex] Factor the numerator. Because the denominator is x + 1, x ≠ - 1. [latex]=\frac{x^2\overset{1}{\cancel{(x + 1)}} }{\underset{1}{\cancel{x + 1}} }[/latex] Divide out the common factor, x + 1. = x², x ≠ -1 Denominators of 1 need not be written because [latex]\frac{a}{1}=a[/latex]. b. [latex]\frac{x^2+6 x+5}{x^2-25}=\frac{(x + 5)(x + 1)}{(x + 5)(x - 5)}[/latex] Factor the numerator and denominator. Because the denominator is (x + 5)(x - 5), x ≠ -5 and x ≠ 5. [latex]=\frac{\overset{1}{\cancel{(x + 5)}} (x + 1)}{\underset{1}{\cancel{(x + 5)}} (x - 5)}[/latex] Divide out the common factor, x + 5. [latex]=\frac{x + 1}{x - 5}[/latex], x ≠ -5, x ≠ 5

Question P.6.2: Simplifying Rational Expressions Simplify: a. x³ + x²/x + 1 ...

Precalculus

Simplifying Rational Expressions

Simplify:

a. $\frac{x^3+x^2}{x+1}$ b. $\frac{x^2+6 x+5}{x^2-25}$ .

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a. $\frac{x^3+x^2}{x+1}=\frac{x^2(x+1)}{x+1}$ Factor the numerator. Because the denominator is x + 1, x ≠ –1.

$=\frac{x^2\overset{1}{\cancel{(x + 1)}} }{\underset{1}{\cancel{x + 1}} }$ Divide out the common factor, x + 1.

= x², x ≠ -1 Denominators of 1 need not be written because $\frac{a}{1}=a$ .

b. $\frac{x^2+6 x+5}{x^2-25}=\frac{(x + 5)(x + 1)}{(x + 5)(x – 5)}$ Factor the numerator and denominator. Because the denominator is (x + 5)(x – 5), x ≠ -5 and x ≠ 5.

$=\frac{\overset{1}{\cancel{(x + 5)}} (x + 1)}{\underset{1}{\cancel{(x + 5)}} (x – 5)}$ Divide out the common factor, x + 5.

$=\frac{x + 1}{x – 5}$ , x ≠ -5, x ≠ 5