Reducing to lowest terms Reduce each rational expression to lowest terms. a. [latex] \frac{2 x+4}{x^{2}+5 x+6}[/latex] b. [latex] \frac{b-a}{a^{3}-b^{3}}[/latex] c. [latex] \frac{x^{2} z^{3}}{x^{5} z}[/latex]

a. [latex] \frac{2 x+4}{x^{2}+5 x+6}=\frac{2(x+2)}{(x+2)(x+3)}[/latex] Factor the numerator and denominator. [latex] =\frac{2}{x+3}[/latex] Divide out the common factor x + 2. b. [latex] \frac{b-a}{a^{3}-b^{3}}=\frac{-1(a-b)}{(a-b)\left(a^{2}+a b+b^{2}\right)}[/latex] Factor -1 out of b - a. [latex] =\frac{-1}{a^{2}+a b+b^{2}}[/latex] c. [latex] \frac{x^{2} z^{3}}{x^{5} z}=\frac{\left(x^{2} z\right)\left(z^{2}\right)}{\left(x^{2} z\right)\left(x^{3}\right)} =\frac{z^{2}}{x^{3}}[/latex] (The quotient rule yields the same result.)

Question B.5.2: Reducing to lowest terms Reduce each rational expression to ...

Precalculus Functions and Graphs

Reducing to lowest terms

Reduce each rational expression to lowest terms.

a. $\frac{2 x+4}{x^{2}+5 x+6}$

b. $\frac{b-a}{a^{3}-b^{3}}$

c. $\frac{x^{2} z^{3}}{x^{5} z}$

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a. $\frac{2 x+4}{x^{2}+5 x+6}=\frac{2(x+2)}{(x+2)(x+3)}$ Factor the numerator and denominator.

$=\frac{2}{x+3}$ Divide out the common factor x + 2.

b. $\frac{b-a}{a^{3}-b^{3}}=\frac{-1(a-b)}{(a-b)\left(a^{2}+a b+b^{2}\right)}$ Factor -1 out of b – a.

=\frac{-1}{a^{2}+a b+b^{2}}

c. $\frac{x^{2} z^{3}}{x^{5} z}=\frac{\left(x^{2} z\right)\left(z^{2}\right)}{\left(x^{2} z\right)\left(x^{3}\right)} =\frac{z^{2}}{x^{3}}$ (The quotient rule yields the same result.)