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Using Factoring in Rational Expressions

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[MUSIC PLAYING] Let's look at our objectives for today. We'll start by reviewing what is the greatest common factor. We'll then introduce rational expressions. And finally, we'll do some examples factoring rational expressions.

Let's review greatest common factors. Terms in a polynomial may have common factors, which are numbers that divide each term in the polynomial. Common terms can be factored out in order to simplify the polynomial. The greatest common factor is the product of all common factors in a polynomial. And factoring out common factors is especially useful when simplifying algebraic fractions or rational expressions.

Now let's look at rational expressions. A rational expression is a fraction whose numerator and denominator are both polynomials. Rational expressions can be simplified in a similar way numeric fractions are. We first identify common factors in the numerator and denominator, and then we cancel out any common factors.

Now let's do some examples factoring rational expressions. We want to factor x squared plus 4x over x squared minus 12x. The terms in the numerator both have an x. So we can factor it out. After taking out an x, the remaining factors go in the parentheses. The remaining factor of x squared is x. And the remaining factor of 4x is 4.

The terms in the denominator also both have an x. So we can factor it out. After taking out an x, the remaining factor of x squared is x. And the remaining factor of negative 12x is negative 12. Now we see that the numerator and denominator both have a common factor of x. And we can cancel out these x terms, because x over x is equal to 1.

So we have x plus 4 over x minus 12. The terms remaining in the numerator and denominator cannot be canceled out, because they are separated by addition or subtraction. Only terms separated by multiplication are canceled out by the division operation of a fraction.

Here's our second example. We want to factor x squared plus 7x plus 10 over x squared plus 4x minus 5. We notice that these are both quadratic expressions in the numerator and the denominator. And we may be able to write these expressions in factored form. To check, in order to factor, we need to find two numbers that multiply to the constant term but add to the coefficient of the x term.

So in the numerator, we need to find two numbers that multiply to 10 an add to 7. These two numbers are 2 and 5. So we can factor our numerator as x plus 2 times x plus 5. In the denominator, we need two numbers that multiply to negative 5 and add to 4. These two numbers are positive 5 and negative 1's. So our denominator factors to x plus 5 times x minus 1.

We now see that we have a common factor in the numerator and denominator of x plus 5. We can cancel out these x plus 5 factors, because x plus 5 over x plus 5 is equal to 1. So our final answer is x plus 2 over x minus 1.

Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. Terms in a polynomial may have common factors, which are numbers that divide each term in the polynomial. Common terms can be factored out in order to simplify the polynomial. A rational expression is a fraction whose numerator and denominator are polynomials. And when canceling out terms in the numerator and denominator, only terms separated by multiplication are canceled out by the division operation of a fraction.

So I hope that these important points and examples helped you understand a little bit more about using factoring in irrational expressions. Keep using your notes, and keep on practicing. And soon you'll be a pro. Thanks for watching.