Certainly! Let's factor out the common terms in the given expression [tex]\(16x^3y - 12x^2y^2 + 32xy^3\)[/tex]. Here is a detailed, step-by-step solution:
1. Identify the Greatest Common Factor (GCF):
- For the coefficients: The coefficients are 16, 12, and 32. The GCF of these numbers is 4.
- For the [tex]\(x\)[/tex] terms: The terms have [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex], and [tex]\(x^1\)[/tex]. The GCF here is [tex]\(x^1 = x\)[/tex].
- For the [tex]\(y\)[/tex] terms: The terms have [tex]\(y\)[/tex], [tex]\(y^2\)[/tex], and [tex]\(y^3\)[/tex]. The GCF here is [tex]\(y^1 = y\)[/tex].
2. Factor out the GCF:
- The overall GCF of the expression is [tex]\(4xy\)[/tex]. We can factor this out from each term of the expression:
[tex]\[
16x^3y - 12x^2y^2 + 32xy^3 = 4xy \cdot \left(\frac{16x^3y}{4xy} - \frac{12x^2y^2}{4xy} + \frac{32xy^3}{4xy}\right)
\][/tex]
3. Simplify Inside the Parentheses:
- Simplify each term inside the parentheses:
[tex]\[
\frac{16x^3y}{4xy} = 4x^2
\][/tex]
[tex]\[
\frac{12x^2y^2}{4xy} = 3xy
\][/tex]
[tex]\[
\frac{32xy^3}{4xy} = 8y^2
\][/tex]
4. Substitute Back the Simplified Terms:
- After simplifying, substitute the terms back into the expression:
[tex]\[
16x^3y - 12x^2y^2 + 32xy^3 = 4xy \cdot (4x^2 - 3xy + 8y^2)
\][/tex]
Therefore, the factored form of [tex]\(16x^3y - 12x^2y^2 + 32xy^3\)[/tex] is:
[tex]\[
4xy \cdot (4x^2 - 3xy + 8y^2)
\][/tex]
