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Chapter 7: Problem 51

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{3}-8}{x-2}$$

Short Answer

Expert verified
The simplified form of the rational expression \(\frac{x^{3}-8}{x-2}\) is \(x^{2} + 2x + 4\).

Step by step solution

01

Factorize the numerator

Rewrite the expression \(x^{3}-8\) as \((x)^3 - (2)^3\). A cube of two terms can be factored using the formula \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\). So, when we apply the formula to the cube difference of our expression, we obtain \((x-2)(x^{2} + 2x + 4)\).
02

Cancel out common term

After the factorization of the original expression we have \(\frac{(x-2)(x^{2} + 2x + 4)}{x-2}\). We can see that \((x-2)\) in the numerator and the denominator are the same. They will cancel each other out when we divide. After that, we are left with \(x^{2} + 2x + 4\) which is already in simplified form.

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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. $$\frac{x^{2}}{x-2}+\frac{4}{2-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{x-1}+\frac{3}{x+2}$$

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