Factor​ completely, or state that the polynomial is prime.
2 x cubed plus x Superscript 6 Baseline minus 24
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Part 1
Select the correct choice below and fill in any answer boxes within your choice.
A.2 x cubed plus x Superscript 6 Baseline minus 24 equals nothing
2 x cubed plus x Superscript 6 Baseline minus 24 equals nothing
  
Your answer is correct.B.
The polynomial is prime.

To factor \(2x^3 + x^6 - 24\), let's go through the following steps:

1. **Rewrite the polynomial in descending order:**  
   \(x^6 + 2x^3 - 24\).

2. **Check for factoring by grouping or substitution:**  
   Notice that \(x^6\) and \(x^3\) terms suggest a possible substitution with \(u = x^3\). Thus, we can rewrite the polynomial in terms of \(u\):
   \[
   u^2 + 2u - 24
   \]

3. **Factor \(u^2 + 2u - 24\):**
   We need two numbers that multiply to \(-24\) and add up to \(2\). These numbers are \(6\) and \(-4\):
   \[
   u^2 + 2u - 24 = (u + 6)(u - 4)
   \]

4. **Substitute back \(u = x^3\):**
   \[
   (x^3 + 6)(x^3 - 4)
   \]

5. **Factor further if possible:**  
   The term \(x^3 - 4\) can be factored as a difference of cubes:
   \[
   x^3 - 4 = x^3 - (2)^3 = (x - 2)(x^2 + 2x + 4)
   \]

   Thus, the complete factorization is:
   \[
   x^6 + 2x^3 - 24 = (x^3 + 6)(x - 2)(x^2 + 2x + 4)
   \]

### Final Answer:
\[
2x^3 + x^6 - 24 = (x^3 + 6)(x - 2)(x^2 + 2x + 4)
\]

Would you like a detailed explanation for each factoring step?

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1. How do you identify common terms in a polynomial for substitution?
2. What are the steps to factor a polynomial by grouping?
3. How can you determine if a polynomial can be factored further?
4. What is the difference of cubes formula?
5. What is the general approach for factoring polynomials with higher powers?

**Tip:** Always start by looking for common factors and substitution opportunities to simplify complex polynomials.

Factor completely, or state that the polynomial is prime. 2x^3 + x^6 - 24

Explore the complete factorization of the polynomial 2x^3 + x^6 - 24 using algebraic techniques such as substitution and the difference of cubes. Learn key factorization steps to express the polynomial as (x^3 + 6)(x - 2)(x^2 + 2x + 4). Suitable for high school students in Grades 10-12.

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