Problem 49 - Introductory Algebra for College Students

Question

Factor each polynomial using the negative of the greatest common factor. $$-12 x^{2}+18$$

Step-by-Step Solution

Answer

The factored polynomial using the negative of the greatest common factor is \(-6(2x^{2} - 3)\).

Step 1: Identify the Greatest Common Factor

The greatest common factor (GCF) of \(-12x^{2}\) and \(18\) is the largest factor that divides both. Here, the factors of -12 are \(±1, ±2, ±3, ±4, ±6,\) and \( ±12\), and the factors of 18 are \(±1, ±2, ±3, ±6, ±9,\) and \(±18\). The largest common factor that appears in both lists is 6.

Step 2: Find the Negative of the GCF

The negative of the greatest common factor is simply \(−6\).

Step 3: Factor the Polynomial

The polynomial \(-12x^{2} + 18\) can be factored as \(-6(2x^{2} - 3)\). The common factor -6 is taken out and each term of \(-12x^{2}+18\) is divided by -6. This leaves \(2x^{2} - 3\) inside the brackets.