Problem 46 - 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. $$27 x^{2} y^{3}-18 x y^{2}+45 x^{2} y$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(32x^{3}y^{2}-24x^{3}y-16x^{2}y\) using the greatest common factor is \(16x^{2}y(2x^{2}y-2x-1)\)

Step 1: Identify GCF

Identify the greatest common factor of all the terms. Observe the powers of \(x\) and \(y\) in each term. The GCF here will be the lowest powers of \(x\) and \(y\) common in all terms including the constant numbers (32, 24, 16). In this case the GCF is \(16x^{2}y\).

Step 2: Factor Out

Factor out the GCF from each term in the expression such that when the obtained expressions are multiplied by the GCF, the original polynomial is obtained. The result will be \(16x^{2}y(2x^{2}y - 2x - 1)\)

Step 3: Check Your Work

Expand the factored expression to ensure that the original polynomial is obtained. Multiplying the GCF by the other expression should give back the original polynomial \((16x^{2}y*2x^{2}y)-(16x^{2}y*2x)+(16x^{2}y*1)\), confirming that it’s correct.