Problem 77 - Introductory Algebra for College Students

Question

Simplify each rational expression. $$\frac{x^{2}-9 x+18}{x^{3}-27}$$

Step-by-Step Solution

Answer

\(\frac{x-6}{x^2+3x+9}\)

Step 1: Factor the Numerator

Rewrite the numerator \(x^{2}-9x+18\) as \((x-3)(x-6)\) using factoring techniques for quadratic expressions.

Step 2: Factor the Denominator

Factoring the denominator involves recognizing the difference of cubes formula, which is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). The denominator \(x^3-27\) can be written as \((x-3)(x^2+3x+9)\)

Step 3: Simplify the Expression

You should cancel any common factors between the numerator and the denominator. In this case, \((x-3)\) appears in both the numerator and the denominator, so it can be cancelled out. This simplifies the expression to \(\frac{x-6}{x^2+3x+9}\).