Question

Factor \(z^{3}+216\).

Short answer

The factored form is \( (z + 6)(z^{2} - 6z + 36) \).

Step-by-step solution

Identify the Cube Terms

Recognize that the given expression is a sum of cubes. Here, we have two cubes: \(z^{3}\) (which is \(z\) cubed) and \(216\) (which is \(6^{3}\)).

Recall the Sum of Cubes Formula

The formula for factoring the sum of cubes is given by \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\).

Assign the Variables

Set \(a = z\) and \(b = 6\). These values come from identifying the cubes in the original expression.

Substitute into the Sum of Cubes Formula

Substitute \(a = z\) and \(b = 6\) into the formula: \( (z + 6)(z^{2} - z \cdot 6 + 6^{2}) \).

Simplify the Expression

Simplify the terms inside the parentheses: \( (z + 6)(z^{2} - 6z + 36) \).

Key concepts

Factoring Polynomials

Factoring polynomials is a key concept in algebra that involves breaking down a polynomial into a product of simpler polynomials. This process is similar to factoring numbers into prime factors. For example, the polynomial expression \(z^{3} + 216\) can be factored into two polynomial factors.