Question
Factor out the greatest common factor. \[ -36 p^{3}+20 p^{2}+12 p \]
Ask by Rowe Ryan. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The greatest common factor is \(4p\), and the expression factored is \(4p(-9p^2 + 5p + 3)\).
Solution
To factor out the greatest common factor (GCF) from the expression:
\[
-36 p^{3} + 20 p^{2} + 12 p
\]
**Step 1: Find the GCF of the coefficients**
The coefficients are \(-36\), \(20\), and \(12\).
- **Factors of 36**: \(2^2 \times 3^2\)
- **Factors of 20**: \(2^2 \times 5\)
- **Factors of 12**: \(2^2 \times 3\)
The common factor among the coefficients is \(2^2 = 4\).
**Step 2: Find the GCF of the variables**
The variable terms are \(p^3\), \(p^2\), and \(p\).
The smallest power of \(p\) common to all terms is \(p\).
**Step 3: Combine the GCFs**
The overall GCF is \(4p\).
**Step 4: Factor out the GCF**
\[
-36 p^{3} + 20 p^{2} + 12 p = 4p \left( -9 p^{2} + 5 p + 3 \right)
\]
So, the factored form is:
\[
4p (-9p^2 + 5p + 3)
\]
**Final Answer:**
\[
4p\,(-9\,p^{2} + 5\,p + 3)
\]
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 \(-36 p^{3}+20 p^{2}+12 p\), we first identify the GCF of the coefficients \(-36\), \(20\), and \(12\). The GCF of these numbers is \(4\). Next, we also consider the variable part, which has the lowest power of \(p\) as \(p\) itself. Therefore, the overall GCF is \(4p\). Now, we can factor \(4p\) out of each term: \[ -36 p^{3}+20 p^{2}+12 p = 4p(-9 p^{2}+5 p+3) \] Thus, the factored form of the expression is: \[ 4p(-9 p^{2}+5 p+3) \]
