To factorize the given expression [tex]\(24 x^2 + 6 x y\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
Start by identifying the greatest common factor of all terms in the polynomial.
The terms are [tex]\(24 x^2\)[/tex] and [tex]\(6 x y\)[/tex]. The numerical coefficients are 24 and 6, and the highest common factor of these numbers is 6. Both terms also share a common variable, [tex]\(x\)[/tex].
So, the GCF of [tex]\(24 x^2\)[/tex] and [tex]\(6 x y\)[/tex] is [tex]\(6x\)[/tex].
2. Factor out the GCF:
Now, divide each term by the GCF [tex]\(6x\)[/tex]:
[tex]\[
= 24 x^2 \div 6x + 6 x y \div 6x
\][/tex]
Simplifying each term:
[tex]\[
= 4x + y
\][/tex]
Therefore, when the GCF [tex]\(6x\)[/tex] is factored out from the original expression, it becomes:
[tex]\[
24 x^2 + 6 x y = 6x(4x + y)
\][/tex]
So, the completely factorized form of the expression [tex]\(24 x^2 + 6 x y\)[/tex] is:
[tex]\[
\boxed{6x(4x + y)}
\][/tex]
