Question
Factor out the greatest common factor. \[ -4 p^{3}+32 p^{2}+12 p \]
Ask by King Gibbs. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
Factor out the greatest common factor \(4p\) from the expression:
\[
-4p(p^2 - 8p - 3)
\]
Solution
To factor out the greatest common factor (GCF) from the expression
\[
-4 p^{3} + 32 p^{2} + 12 p,
\]
follow these steps:
1. **Identify the GCF of the coefficients:**
- The coefficients are \(-4\), \(32\), and \(12\).
- The factors of \(4\) are \(1, 2, 4\).
- The factors of \(32\) are \(1, 2, 4, 8, 16, 32\).
- The factors of \(12\) are \(1, 2, 3, 4, 6, 12\).
- The common factors are \(1, 2, 4\), so the GCF of the coefficients is \(4\).
2. **Identify the GCF of the variables:**
- The variables are \(p^3\), \(p^2\), and \(p\).
- The smallest exponent of \(p\) is \(1\), so the GCF of the variables is \(p\).
3. **Combine the GCF of the coefficients and variables:**
- The overall GCF is \(4p\).
4. **Factor out the GCF from each term:**
\[
-4 p^{3} + 32 p^{2} + 12 p = -4p(p^{2} - 8p - 3)
\]
So, the expression factored by the greatest common factor is:
\[
-4p(p^{2} - 8p - 3)
\]
**Final Answer:**
\[
-4\,p\left(p^{2} - 8\,p\,-\,3\,\right)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To factor out the greatest common factor (GCF) from the expression \(-4p^{3} + 32p^{2} + 12p\), first identify the GCF of the coefficients -4, 32, and 12. The GCF here is 4, and since all terms also include \(p\), we can factor out \(4p\). Now, rewriting the expression gives us: \[ -4p(p^{2} - 8p - 3) \] Next, let's check if the quadratic \(p^{2} - 8p - 3\) can be factored further. Since the discriminant (\(b^2 - 4ac\)) is not a perfect square, it doesn’t have rational factors, and it remains as is. So, the factored form is: \[ -4p(p^{2} - 8p - 3) \] And that’s how you factor out the GCF like a pro!
