# Null Factor Law

I’d like to return to algebra and discuss the Null Factor Law which is useful in solving certain equations:

If two or more factors multiplied together equal zero, then the solutions can be found be equating each factor separately to zero.

This makes sense if you think of two numbers multiplied together equal 0:

${ab}\hspace{0.33em}{=}\hspace{0.33em}{0}$ can only be true if either a is zero, b is zero, or both are zero. No other non-zero numbers multiplied together can equal zero.

This is true for any algebraic expressions multiplied together. For example:

${(}{x}{-}{7}{)(}{x}{+}{5}{)}\hspace{0.33em}{=}\hspace{0.33em}{0}$

Can only be true if ${(}{x}{-}{7}{)\hspace{0.33em}=}\hspace{0.33em}{0}$ or if ${(}{x}{+}{5}{)\hspace{0.33em}=}\hspace{0.33em}{0}$

Without formal algebra, you can see the two solutions to this equation are then x = 7 or -5.

Now this one was easy, but sometimes you are given an equation that is not a factored one:

${x}^{2}\hspace{0.33em}{+}\hspace{0.33em}{5}{x}\hspace{0.33em}{=}\hspace{0.33em}{0}$

At first glance, it looks like the null factor law doesn’t apply here. But I did a post on the distributive property. Please review that if needed, but notice that there is a common factor of x in each of the terms on the left side of the equation. I can un-distribute this x to get:

${x}{(}{x}\hspace{0.33em}{+}\hspace{0.33em}{5}{)}\hspace{0.33em}{=}\hspace{0.33em}{0}$

Looks like the null factor law can be used as there are now two factors on the left side. So mentally setting each of these to zero, we get the two solutions x = 0 or -5.

I covered a similar example on my post about quadratic equations. You can click on the tags on the right or below (depending on the device you are viewing this on) to directly go to previous posts on the listed topics.

I will be covering more complex examples in my next post.