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Solving by Factoring - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

The first step is to bring all the terms to one side and set the equation equal to zero. Next, using the method of solving by factoring, take out the common terms and use one of the methods of factoring to simplify the expression. Once in this multiplication form, note that if two terms multiplied equal zero, one of the terms must be equal to zero. Given the rational roots theorem, these are the solutions to the equation.

Solving a polynomial by factoring. So this particular problem what we're going to talk about is when we have any sort of polynomial, any sort of equation that deals with the multiple degrees of x's and we're trying to solve it out.
Whenever we have this kind of thing, we always have to bring everything to one side. Okay? So it's easiest to compare something to zero. In general we're used to having our largest term have a positive coefficient so in this case I want to bring the -6 around and we end up with x squared minus 7x plus 6 is equal to zero.
Now it's really hard when we have an equation and we're adding and subtracting things and setting it equal to another item because we don't really know what each of these could be in order to add up to zero. But what we can do is turn it into a multiplication problem and we know how to do that by factoring, okay? So we can easily factor this. x-6, x-1 is equal to zero. And now where the multiplication and the fact it's equal to zero comes in handy is that we have two things being multiplied equal to zero, okay? So what that tells us is we have two things multiplied equal zero, one of them has to be zero. So what that tells us is x either has to be 6 making this zero, or x has to be 1 to make this zero, okay? As long as one of these terms is zero, you multiply them together, equal zero as well.
So whenever solving a polynomial, bring everything to one side, factor it out and then knowing that two things being multiplied equals zero, one of them has to be equals zero. It's really easy to solve this out.

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