Problem 44 - 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. $$4 x^{2} y^{3}+6 x y$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(4 x^{2} y^{3}+6 x y\) is \(2xy (2x^{2}y^{2} + 3 )\).

Step 1: Identify the Greatest Common Factor

The first step in factoring a polynomial is to identify the GCF of the entire expression. In the given expression \(4 x^{2} y^{3}+6 x y\), the GCF is the product of the highest powers of common factors present in both terms. Here, the common factors are 2, x and y. So, the GCF is obtained by multiplying these common factors, that gives \(2xy\).

Step 2: Factor out the GCF

Next, the GCF identified in Step 1 is factored out from each term of the polynomial. Express each term as the product of the GCF and the remaining factors. The given expression \(4 x^{2} y^{3}+6 x y\) becomes \(2xy (2x^{2}y^{2} + 3 )\) .

Step 3: Simplify the expression

When the polynomial is factored out and expressed as a product, it is important to simplify any possible further terms . Our factored expression \(2xy (2x^{2}y^{2} + 3 )\) cannot be simplified any further.