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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q. 7.32 (page 659)

(a) Find the LCD for the expressions $\frac{3 x}{x^{2} - 3 x + 10}, \frac{5}{x^{2} + 3 x + 2}$
(b) rewrite them as equivalent rational expression
with the lowest common denominator.

Short Answer

Expert verified

(a) The LCD is $(x - 5) (x + 2) (x + 1)$

(b) The required expressions are $\frac{3 x (x + 1)}{(x - 5) (x + 2) (x + 1)}, \frac{5 (x - 5)}{(x - 5) (x + 2) (x + 1)}$

Step by step solution

01

Part (a) Step 1. Given information

The given expressions are $\frac{3 x}{x^{2} - 3 x + 10}, \frac{5}{x^{2} + 3 x + 2}$

02

Part (a) Step 2. Find LCD

Factor the denominators of the expressions.

$\begin{array}{rcl} x^{2} - 3 x + 10 & = & (x - 5) (x + 2) \\ x^{2} + 3 x + 2 & = & (x + 1) (x + 2) \end{array}$

So, the LCD will be $(x - 5) (x + 2) (x + 1)$ .

03

Part (b) Step 1. Given information

The given expressions are $\frac{3 x}{x^{2} - 3 x + 10}, \frac{5}{x^{2} + 3 x + 2}$

04

Part (b) Step 2. Multiply by missing Factors

Multiply each rational expressions' numerator and denominator by the missing factor.

$\frac{3 x}{x^{2} - 3 x + 10} = \frac{3 x (x + 1)}{(x - 5) (x + 2) (x + 1)} \frac{5}{x^{2} + 3 x + 2} = \frac{5 (x - 5)}{(x - 5) (x + 2) (x + 1)}$

So, the rational expressions with common denominators are $\frac{3 x (x + 1)}{(x - 5) (x + 2) (x + 1)}, \frac{5 (x - 5)}{(x - 5) (x + 2) (x + 1)}$ .

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