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Chapter 7: Problem 52

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{3}-125}{x^{2}-25}$$

Short Answer

Expert verified
The simplified form of the rational expression is \(\frac{x^{2}+5x+25}{x+5}\).

Step by step solution

01

Identify the patterns

Identify that both the numerator and the denominator have a difference of cubes and squares pattern. \[x^{3}-125\] is a difference of cubes and \[x^{2}-25\] is a difference of squares.
02

Factor the numerator and the denominator

Factor the numerator using the difference of cubes formula \(a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})\) where \(a=x\) and \(b=5\). This results in \((x-5)(x^{2}+5x+25)\). Factor the denominator using the difference of squares formula \(c^{2}-d^{2}=(c+d)(c-d)\), where \(c=x\) and \(d=5\). Resulting to \((x+5)(x-5)\). The given expression becomes \[\frac{(x-5)(x^{2}+5x+25)}{(x+5)(x-5)}\].
03

Cancel common factors

Notice that the common factor of \(x-5\) can be cancelled from both the numerator and the denominator, resulting in the final simplified expression of \[\frac{x^{2}+5x+25}{x+5}.\]

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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. $$\frac{6 x+7}{x-6}+\frac{3 x}{6-x}$$

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