To factor the given polynomial [tex]\( 9x^4y - 6x^3y^2 + 3x^2y^3 \)[/tex], you should follow these steps:
### Step 1: Identify the Greatest Common Factor (GCF)
First, look for the greatest common factor in each term. The terms are:
- [tex]\( 9x^4y \)[/tex]
- [tex]\( 6x^3y^2 \)[/tex]
- [tex]\( 3x^2y^3 \)[/tex]
The GCF of the numerical coefficients (9, 6, and 3) is 3. For the variables, the smallest power of [tex]\( x \)[/tex] present in each term is [tex]\( x^2 \)[/tex], and the smallest power of [tex]\( y \)[/tex] is [tex]\( y \)[/tex]. Therefore, the GCF is [tex]\( 3x^2y \)[/tex].
### Step 2: Factor out the GCF
Divide each term by the GCF ([tex]\( 3x^2y \)[/tex]) and rewrite the polynomial in this factored form:
[tex]\[
9x^4y - 6x^3y^2 + 3x^2y^3 = 3x^2y \left(\frac{9x^4y}{3x^2y} - \frac{6x^3y^2}{3x^2y} + \frac{3x^2y^3}{3x^2y}\right)
\][/tex]
Simplify each term within the parentheses:
- [tex]\(\frac{9x^4y}{3x^2y} = 3x^2\)[/tex]
- [tex]\(\frac{6x^3y^2}{3x^2y} = 2xy\)[/tex]
- [tex]\(\frac{3x^2y^3}{3x^2y} = y^2\)[/tex]
### Step 3: Rewrite the Polynomial
Substitute these simplified terms back into the expression:
[tex]\[
3x^2y \left(3x^2 - 2xy + y^2\right)
\][/tex]
Thus, the factored form of the given polynomial [tex]\( 9x^4y - 6x^3y^2 + 3x^2y^3 \)[/tex] is:
[tex]\[
3x^2y \left(3x^2 - 2xy + y^2\right)
\][/tex]
### Verify the Factored Form
To ensure correctness, you can expand the factored form:
[tex]\[
3x^2y \left(3x^2 - 2xy + y^2\right) = 3x^2y \cdot 3x^2 - 3x^2y \cdot 2xy + 3x^2y \cdot y^2
\][/tex]
[tex]\[
= 9x^4y - 6x^3y^2 + 3x^2y^3
\][/tex]
This matches the original polynomial, confirming that the factorization is correct.
### Conclusion
Hence, the correct factored form is:
[tex]\[
3x^2y \left(3x^2 - 2xy + y^2\right)
\][/tex]
The correct answer is:
[tex]\[
3x^2 y\left(3x^2 - 2xy + y^2\right)
\][/tex]
