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Chapter 7: Problem 15

Find the least common denominator of the rational expressions. $$\frac{3}{x^{2}-x-20} \text { and } \frac{x}{2 x^{2}+7 x-4}$$

Short Answer

Expert verified
The least common denominator of the given rational expressions is \((x+4)*(x-5)*(2x-1)\)

Step by step solution

01

Factorise the denominators

The first step is to factorise the denominators. The factors of \(x^{2}-x-20\) are \((x+4)\) and \((x-5)\), because \(-5 * 4 = -20\) and \(-5 + 4 = -1\). Similarly, the factors of \(2x^2+7x-4\) are \((2x-1)\) and \((x+4)\), because \(1*4* 2 = -8\) and \(-8 + 7 = -1\).
02

Identify the common factors

Having factored the denominators, we can now identify the common factors. From the factorised denominators, it is clear that \((x+4)\) is the common factor.
03

Find the least common denominator

The least common denominator is obtained by multiplying the unique factors from both expressions. Thus, the least common denominator for the given rational expressions is \((x+4)*(x-5)*(2x-1)\).

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Most popular questions from this chapter

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{4}{x-3}+\frac{2}{3-x}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 y}{x^{2}-y^{2}}+\frac{2 x}{y^{2}-x^{2}}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{4-x}{x-9}-\frac{3 x-8}{9-x}$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27}$$
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