Problem 81 - Introductory Algebra for College Students

Question

Factor using the formula for the sum or difference of two cubes. $$x^{3}-27$$

Step-by-Step Solution

Answer

The factorized form of \(x^3 - 27\) is \((x - 3)(x^2 + 3x + 9)\).

Step 1: Identifying terms

We can identify \(a\) and \(b\) by matching the given equation with the formula for difference of cubes. Here, \(a = x\) and \(b = 3\) because \(x^3\) is the cube of \(x\) and \(27\) is the cube of \(3\).

Step 2: Substituting the values

Substitute the values of \(a\) and \(b\) in the the formula for the difference of two cubes which is \((a - b)(a^2 + ab + b^2)\). Therefore, the factored form of the equation is \((x - 3)(x^2 + 3x + 9)\).