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Chapter 7: Problem 49

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

Short Answer

Expert verified
The simplified form of the given rational expression is \(x^{2}+k\), where \(k\) is a constant.

Step by step solution

01

Check if the denominator is a factor of the numerator

To simplify a fraction, a useful method is to factorize the numerator and the denominator and then cancel out common factors. Therefore, the polynomial \(x^{3}-2 x^{2}+x-2\)needs to be checked if it can be factored with (x-2) as a factor.
02

Factorization

To factorize the given polynomial \(x^{3}-2 x^{2}+x-2\), apply the factor theorem, which states that if a polynomial \(f(x)\) gives a result of zero when \(x = a\), then \((x-a)\) is a factor of that polynomial. So, plug \(x=2\) into the polynomial and if the result is zero, then \(x-2\) is its factor. After calculation, \(2^{3} - 2*2^{2} + 2 - 2 = 8 - 8 + 2 - 2 = 0\), so \(x-2\) is a factor of \(x^{3}-2 x^{2}+x-2\). Additionally, the polynomial can be reduced to \(x^{3}-2 x^{2}+x-2 = (x-2)(x^{2}+k)\), where \(k\) is a constant.
03

Simplify

Now, the original rational expression \(\frac{x^{3}-2 x^{2}+x-2}{x-2}\) can be simplified to \(\frac{(x-2)(x^{2}+k)}{(x-2)}\). Simplify the result further by cancelling out the common factors, which leaves the rational expression \(x^{2}+k\) as the simplest form.

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Most popular questions from this chapter

Graph: \(y=-\frac{2}{3} x+4 .\) (Section 3.4, Example 3)

Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)

Solve: \(x^{2}-12 x+36=0 .\) (Section 6.6, Example 4)

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

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