Problem 49 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.