Problem 43 - Introductory Algebra for College Students

Question

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state. $$6 x^{3} y^{2}+9 x y$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(6x^{3}y^{2} + 9xy\) is \(3xy(2x^{2}y + 3)\)

Step 1: Determine the GCF

In the polynomial \(6x^{3}y^{2}+9xy\), first determine the greatest common factor of the coefficients and the variables. The GCF of 6 and 9 is 3; the smallest power for \(x\) is 1 and \(y\) is also 1. So the GCF is \(3xy\).

Step 2: Divide each term by the GCF

Now, divide each term of the polynomial by the GCF. This gives: \((6x^{3}y^{2} ÷ 3xy) + (9xy ÷ 3xy)\).

Step 3: Simplify

Simplify equation, which yields \(2x^{2}y + 3\).

Step 4: Write the factored form

Combine the GCF and the simplified polynomial to get the factored form. So, \(6x^{3}y^{2}+9xy = 3xy(2x^{2}y + 3)\)