Problem 84 - Introductory Algebra for College Students

Question

Simplify each rational expression. $$\frac{x^{3}-3 x^{2}+9 x}{x^{3}+27}$$

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{x}{x+3}\)

Step 1: Factorize the numerator

To factorize the numerator, the common term \(x\) is taken out: \(x^{3}-3 x^{2}+9 x= x(x^{2}-3 x+9)\)

Step 2: Factorize the denominator

For the denominator, recognise that \(x^{3}+27\) can be written as a sum of cubes: \(x^{3}+27 = x^{3} + 3^3\). Hence, using the formula \(a^{3}+b^{3}= (a+b)(a^{2}-ab+b^{2})\), we get \(x^{3} + 27 = (x+3)(x^{2}-3x+9)\).

Step 3: Cancel out common factors

Now the expression becomes \(\frac{x(x^{2}-3 x+9)}{(x+3)(x^{2}-3x+9)}\). The term \(x^{2} - 3x + 9\) in the numerator and the denominator cancels out, therefore the simplified version would be \(\frac{x}{x + 3}\)