The expression \( x^6 + 4x^3 - 5 \) is a polynomial. If you're looking to simplify, factor, or solve this polynomial, here's a breakdown of what can be done:

### 1. Factoring:
This is a degree-6 polynomial, and it looks like it might be factorable using substitution. Let’s use the substitution:

Let \( u = x^3 \), which means that \( x^6 = u^2 \). The expression then becomes:
\[
u^2 + 4u - 5
\]
This is now a quadratic equation in terms of \( u \), which can be factored as:
\[
(u + 5)(u - 1)
\]
Now, substitute \( u = x^3 \) back into the factored form:
\[
(x^3 + 5)(x^3 - 1)
\]
The second factor, \( x^3 - 1 \), can be factored further using the difference of cubes formula:
\[
x^3 - 1 = (x - 1)(x^2 + x + 1)
\]
So the fully factored form of the original polynomial is:
\[
(x^3 + 5)(x - 1)(x^2 + x + 1)
\]

### Final factored form:
\[
(x^3 + 5)(x - 1)(x^2 + x + 1)
\]

Would you like a more detailed explanation of the steps or a specific method like solving for roots?

---

Here are 5 related questions to extend your understanding:
1. How do we solve for the roots of the factored form \( (x^3 + 5)(x - 1)(x^2 + x + 1) = 0 \)?
2. What is the difference of cubes formula, and how is it derived?
3. How would this expression change if an additional constant were added or subtracted?
4. What are some real-world applications of solving polynomials like this?
5. How would we graph \( x^6 + 4x^3 - 5 \) and analyze its behavior?

**Tip:** Always check for possible substitutions when factoring higher-degree polynomials, as it can simplify the process dramatically!

Explore the step-by-step factoring and solving of the polynomial x^6 + 4x^3 - 5. This guide covers key algebraic concepts like substitution and the difference of cubes formula, providing a detailed breakdown suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation