Problem 82 - Introductory Algebra for College Students

Question

Factor using the formula for the sum or difference of two cubes. $$x^{3}-64$$

Step-by-Step Solution

Answer

The factored form of \(x^3 - 64\) is \((x - 4)(x^2 + 4x + 16)\)

Step 1: Identify a and b

In the equation \(x^3 - 64\), you can simplify it to be read as \(x^3 - 4^3\) where \(x^3\) is \(a^3\) and \(4^3\) is \(b^3\). That makes \(a = x\) and \(b = 4\).

Step 2: Apply the Difference of Cubes Formula

Substitute \(a\) and \(b\) into the difference of cubes formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). So it becomes \(x^3 - 4^3 = (x - 4)(x^2 + 4x + 4^2)\).

Step 3: Simplify the expression

Further simplify the expression to obtain \(x^3 - 64 = (x - 4)(x^2 + 4x + 16)\). This is the factored form of the given expression.