Question
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Answer
The expression \( 12 p^{3} - 75 p x^{2} \) factors to \( 3p (2p - 5x)(2p + 5x) \).
Solution
To factor the expression \( 12 p^{3} - 75 p x^{2} \), follow these steps:
1. **Identify the Greatest Common Factor (GCF):**
- The GCF of \( 12 p^{3} \) and \( 75 p x^{2} \) is \( 3p \).
2. **Factor out the GCF:**
\[
12 p^{3} - 75 p x^{2} = 3p (4 p^{2} - 25 x^{2})
\]
3. **Recognize the Difference of Squares:**
- The expression inside the parentheses, \( 4 p^{2} - 25 x^{2} \), is a difference of squares and can be factored further:
\[
4 p^{2} - 25 x^{2} = (2p)^{2} - (5x)^{2} = (2p - 5x)(2p + 5x)
\]
4. **Combine the Factors:**
\[
12 p^{3} - 75 p x^{2} = 3p (2p - 5x)(2p + 5x)
\]
**Final Factored Form:**
\[
3p (2p - 5x)(2p + 5x)
\]
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Simplify this solution Extra Insights
To factor out the expression \( 12p^3 - 75px^2 \), first, we identify the greatest common factor (GCF) of the coefficients and the variables. The GCF here is \( 3p \). So, we can factor it out: \( 12p^3 - 75px^2 = 3p(4p^2 - 25x^2) \). Now, notice that \( 4p^2 - 25x^2 \) is a difference of squares, which can be further factored: \( 4p^2 - 25x^2 = (2p - 5x)(2p + 5x) \). Putting it all together, the fully factored form of the expression is: \( 12p^3 - 75px^2 = 3p(2p - 5x)(2p + 5x) \).
