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Factor the GCF from the expression: [tex]12x^3y + 6x^2y^2 - 9xy^3[/tex].

A. [tex]3xy\left(4x^2 + 2xy - 3y^2\right)[/tex]

B. [tex]3x^2y\left(4x + 2xy - 3\right)[/tex]

C. [tex]3xy\left(4x^2 + 2xy - 3y^2\right)[/tex]

D. [tex]3x^2y\left(4x^3y - 2x^2y^2 - 3xy^3\right)[/tex]

Sagot :

GinnyAnsweridn
Certainly! Let's factor the given expression step-by-step:

Given expression:
[tex]\[12 x^3 y + 6 x^2 y^2 - 9 x y^3\][/tex]

Step 1: Identify the Greatest Common Factor (GCF) of the terms in the expression.

Each term in the expression has the factors:
1. [tex]\(12 x^3 y\)[/tex]: has factors [tex]\(3, 4, x^3, y\)[/tex]
2. [tex]\(6 x^2 y^2\)[/tex]: has factors [tex]\(3, 2, x^2, y^2\)[/tex]
3. [tex]\(9 x y^3\)[/tex]: has factors [tex]\(3, 3, x, y^3\)[/tex]

The common factors are [tex]\(3, x, y\)[/tex].

Combining these, the GCF of the three terms is:
[tex]\[3xy\][/tex]

Step 2: Factor out the GCF from each term in the expression:

[tex]\[ 12 x^3 y + 6 x^2 y^2 - 9 x y^3 = 3xy \left( \frac{12 x^3 y}{3xy} + \frac{6 x^2 y^2}{3xy} - \frac{9 x y^3}{3xy} \right) \][/tex]

Step 3: Perform the division inside the parentheses:

[tex]\[ 3xy \left(4 x^2 + 2 x y - 3 y^2\right) \][/tex]

So, the factored form of the expression is:
[tex]\[3xy \left(4 x^2 + 2 x y - 3 y^2\right)\][/tex]

Therefore, the correct factorization of the given expression [tex]\(12 x^3 y + 6 x^2 y^2 - 9 x y^3\)[/tex] is:

[tex]\[3xy \left(4 x^2 + 2 x y - 3 y^2\right)\][/tex]

Among the given options, the correct answer is:

[tex]\[3 x y\left(4 x^2+2 x y-3 y^2\right)\][/tex]
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