[MUSIC PLAYING] Let's look at our objectives for today. We'll start by introducing rational expressions. We'll then review how to simplify fractions. We'll also review the greatest common factor in polynomials. And finally, we'll do some examples simplifying rational expressions.
Let's start by looking at rational expressions. A rational expression is a fraction whose numerator and denominator are polynomials. They are sometimes referred to as algebraic fractions. Reducing rational expressions is similar to reducing numerical fractions. Factors can be canceled only if they appear as factors of both the numerator and the denominator.
Factors are canceled because they reduce to 1. For example, x plus 3 over x plus 3 simplifies to 1. However, terms separated by addition or subtraction in the numerator or denominator cannot be canceled. For example x minus 2 over x plus 5 does not equal negative 2 over 5. For example, if we substituted a value 3 in for both x's, we'd have 3 minus 2 over 3 plus 5, which simplifies to 1 over 8, again, not negative 2 over 5.
Now let's review how to simplify fractions by canceling common factors in the numerator and denominator. We want to simplify the fraction 48/90. The first step is to write the numerator and denominator as products of prime factors. So we write the numerator, 48, as 2 times 2 times 2 times 2 times 3. And we write the denominator, 90, as 2 times 3 times 3 times 5.
The second step is to cancel factors that appear in both the numerator and denominator. We can cancel out one 2 and one 3 in both the numerator and denominator. The third step is to multiply remaining factors in the numerator and remaining factors in the denominator. In the numerator we have three 2's remaining, 2 times 2 times 2. And in the denominator, we have one 3 and one 5, so 3 times 5. This simplifies to 8/15. Simplifying rational expressions is similar to reducing numerical fractions, because we identify common factors to cancel.
Now let's look at an example of how to find the greatest common factor of a polynomial. We want to factor the expression 4x to the third minus 8x. We can write each term as a product of factors. 4x to the third is 2 times 2 times x times x times x. Negative 8x is negative 2 times 2 times 2 times x.
We see that both terms have two 2's and one x in common. Multiplying these common factors, 2 times 2 times x gives us 4x. So we begin to rewrite our expression by writing our greatest common factor, 4x, on the outside of the parentheses, and writing the remaining factors of each term inside the parentheses. From the first term, we have two x's remaining, or x squared. And in our second term, we have a negative and a 2, so minus 2. So the factored form of our expression is 4x times x squared minus 2. This strategy can sometimes be helpful when simplifying algebraic fractions where common factors appear in the numerator and denominator of a fraction.
Now let's do some examples simplifying rational expressions. We want to simplify 12a to the sixth over 8a to the fourth. We start by writing both the numerator and denominator as a product of its prime factors and variable factors. So 12a to the sixth can be rewritten as 2 times 2 times 3 times a times a times a times a times a times a, six a's multiplied together.
8a to the fourth can be rewritten as 2 times 2 times 2 times a times a times a times a, four a's multiplied together. We then see that both the numerator and denominator have two 2's as common factors, and four a's as common factors. We can cancel all these factors out. This leaves us with three and two a's multiplied together in the numerator, and one 2 in the denominator. Multiplying our remaining factors gives us 3a squared over 2.
Here's our next example. We want to simplify the expression 7x plus 21 over 2x plus 6. We start by factoring our numerator. 7x is 7 times x, and 21 is 7 times 3. We then factor our denominator. 2x is 2 times x, and 6 is 2 times 3. Looking again at our numerator, we see that we have a common factor of 7. So we can factor out a 7 and rewrite our expression with the remaining factors, x and plus 3. So our numerator is 7 times x plus 3.
Looking at our denominator, we see that we have a common factor of 2 in both terms, so we can factor out the 2 by writing it on the outside of the parentheses and writing our remaining factors, x and plus 3. So our denominator is 2 times x plus 3. Now we see that we have a common factor of x plus 3 in both the numerator and the denominator, and it can be canceled out. This leaves us with 7/2.
Let's go over our important points from today. Make sure you get them in your notes so you can refer to them later. A rational expression is a fraction whose numerator and denominator are polynomials. They are sometimes referred to as algebraic fractions. Reducing rational expressions is similar to reducing numerical fractions. Factors can be canceled only if they appear as factors of both the numerator and denominator.
The steps to simplify numerical and algebraic fractions are, step 1, write the numerator and denominator as products of prime factors. Step 2, cancel factors that appear in both the numerator and denominator. And step 3, multiply remaining factors in the numerator and denominator.
So I hope that these important points and examples helped you understand a little bit more about canceling common factors in algebraic fractions. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.
