Certainly! Let's factor the expression completely step-by-step:
Given expression:
[tex]\[ 18 x^4 y^2 + 21 x^5 y - 6 x^3 y^2 \][/tex]
Step 1: Identify the Greatest Common Factor (GCF)
Firstly, we look for the greatest common factor in all the terms.
Term 1: [tex]\(18 x^4 y^2\)[/tex]
Term 2: [tex]\(21 x^5 y\)[/tex]
Term 3: [tex]\(6 x^3 y^2\)[/tex]
The greatest common factor for the coefficients [tex]\(18, 21,\)[/tex] and [tex]\(6\)[/tex] is [tex]\(3\)[/tex].
For the variables [tex]\(x^4, x^5,\)[/tex] and [tex]\(x^3\)[/tex], the greatest common factor is [tex]\(x^3\)[/tex] (the smallest power of [tex]\(x\)[/tex]).
For the variables [tex]\(y^2, y,\)[/tex] and [tex]\(y^2\)[/tex], the greatest common factor is [tex]\(y\)[/tex] (the smallest power of [tex]\(y\)[/tex]).
Therefore, the GCF of the entire expression is [tex]\(3x^3y\)[/tex].
Step 2: Factor out the GCF
We factor [tex]\(3x^3y\)[/tex] out of each term in the expression:
[tex]\[ 3x^3y ( \frac{18 x^4 y^2}{3x^3y} + \frac{21 x^5 y}{3x^3y} - \frac{6 x^3 y^2}{3x^3y} ) \][/tex]
Now, we simplify the terms inside the parentheses:
[tex]\[ = 3x^3y ( 6x y + 7x^2 - 2y ) \][/tex]
Step 3: Write the factored expression
So, the completely factored form of the given expression is:
[tex]\[ 3x^3y ( 7x^2 + 6xy - 2y ) \][/tex]
This is the factored form of the given polynomial:
[tex]\[ 18 x^4 y^2 + 21 x^5 y - 6 x^3 y^2 = 3x^3 y (7x^2 + 6xy - 2y) \][/tex]
