The given function is:

\[
f(x) = x^3 - 64
\]

This is a cubic function. The expression \(x^3 - 64\) can be factored as a difference of cubes, which follows the general formula:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here, \(a = x\) and \(b = 4\), since \(64 = 4^3\). Applying the formula:

\[
f(x) = (x - 4)(x^2 + 4x + 16)
\]

### Key Points:
1. The factorization of the cubic function is \(f(x) = (x - 4)(x^2 + 4x + 16)\).
2. The function has one real root, \(x = 4\).
3. The quadratic factor, \(x^2 + 4x + 16\), does not have real roots, as its discriminant \((b^2 - 4ac = 16 - 64 = -48)\) is negative. Therefore, the other two roots are complex.

Would you like more details on solving or graphing this function? 

Here are 5 related questions:
1. How do we find complex roots of quadratic functions?
2. What is the discriminant and its role in finding roots?
3. How do we factor other types of polynomials?
4. What are the real and complex roots of the equation \(x^2 + 4x + 16 = 0\)?
5. How does the graph of \(f(x) = x^3 - 64\) behave?

**Tip:** Factoring cubic polynomials often requires identifying them as a sum or difference of cubes to simplify the function.

Learn how to factor the cubic function f(x) = x³ - 64 using the difference of cubes formula. This solution breaks down key algebraic concepts, factoring techniques, and the discriminant for complex roots. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation