Problem 54 - Introductory Algebra for College Students

Question

Factor each polynomial using the negative of the greatest common factor. $$-9 a^{2} b^{3}+12 a b$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(-9a^{2}b^{3}+12ab\) using the negative of the GCF is \(-3ab(3b^{2}-4)\).

Step 1: Identify the GCF

First, identify the GCF of the terms \(-9a^{2}b^{3}\) and \(12ab\). The GCF in this case will include both numerical and variable portions. The numerical GCF of 9 and 12 is 3, and the variable portion is \(a\) (lowest power among the two) and \(b\) (lowest power among the two). Therefore, the GCF of \(-9a^{2}b^{3}\) and \(12ab\) is \(3ab\). The negative of the GCF is \(-3ab\).

Step 2: Factor out the negative GCF

Factor out \(-3ab\) from each term in the given polynomial. When we divide \(-9a^{2}b^{3}\) by \(-3ab\), we get \(3ab^{2}\). When we divide \(12ab\) by \(-3ab\), we get \(-4\). Thus, the factored form of the polynomial is \(-3ab(3b^{2}-4)\).