Problem 22 - 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. $$x^{2}+6 x$$

Step-by-Step Solution

Answer

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

Step 1: Identify Common Factors

Looking at the terms in this polynomial, \(x^{2}\) and \(6 x\), both terms have a common factor of \(x\).

Step 2: Factor out the Common Factors

Use the distributive property, also known as factoring out, to separate the common factor. This means \(x^{2}+6 x\) equals \(x(x+6)\). We are essentially reversing the original distribution of \(x\) throughout \(x + 6\). It's just like saying that, 2*(3+4) is the same as 2*3 + 2*4. So, we are doing the same thing but in reverse.

Step 3: Check Your Work

To ensure that the original polynomial was factored correctly, distribute \(x\) throughout \(x + 6\). This should result in the original polynomial \(x^{2}+6 x\).