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Multiplying and Dividing Rational Expressions

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Today we're going to talk about multiplying and dividing rational expressions. So we'll start by reviewing how to multiply and divide numerical fractions. And then we'll do some examples multiplying and dividing rational expressions.

So let's start by reviewing how to multiply and divide numerical fractions. For my first example, I've got 2/7 multiplied by 3/4. When we're multiplying fractions, we multiply numerator by numerator and denominator by denominator.

So for my numerator, I'm going to multiply 2 times 3, which will give me 6. And in my denominator 7 times 4 will give me 28. 6 over 28 can be reduced, because they both have a common factor of 2.

So 6 is the same as 2 times 3. And 28 is the same as 2 times 14. So both 2's will cancel out. And I'm left with 3 over 14. So my answer in most simplified form is 3/14.

Let's try this. 3/8 divided by 1/5. Now, when we're dividing fractions, we simply multiply by the reciprocal of the second fraction. So dividing by 1/5 is the same as multiplying by 5 over 1. So 3 over 8 divided by 1/5 is 3 over 8 times 5 over 1.

Again, when we're multiplying fractions, we simply multiply across in the numerator and in the denominator. So 3 times 5 will give me 15. And 8 times 1 will give me 8. Now, 15 and 8 don't have any common factors, so this fraction cannot be simplified or reduced. So 15 over 8 is my final answer.

So now let's do an example multiplying rational expressions. I've got 8x minus 1 over 3x, multiplied by x squared over x plus 4. Multiplying and dividing rational expressions follows the same procedure as multiplying and dividing numerical fractions. It's just that our numerator and denominator may contain factors, terms, or variables.

So we're going to multiply these fractions together by multiplying numerator by numerator and denominator by denominator. So we're going to start in the numerators. When I multiply 8x minus 1 by x squared, I need to make sure that I multiply both terms in this first fraction by the term in my second fraction, so I'm going to be distributing.

So I'm going to be multiplying 8x times x squared, which is going to give me 8x to the third. And then I'll multiply 8x-- I'm sorry. Then I'll multiply negative 1 by x squared, which will give me a minus x squared.

Same thing in my denominators. I'm going to multiply 3x by both x and the 4. So 3x times x will give me 3x squared. And 3x times 4 would give me plus 4-- sorry-- plus 12x.

All right. So now that I've multiplied by two fractions together, I can see if I can simplify. And I can, because each of these terms in the numerator and the denominator have a common factor of x. So I'm going to factor out an x both in the terms in the numerator and in the denominator.

So in my numerator, when I factor out an x, I'll have 8x squared. x times 8x squared will give me 8x to the third. And then here I'll have minus x, because x times minus x will give me a minus or negative x squared.

In the denominator, I'll also factor out an x. And then I'll have left 3x plus 12. x times 3x gives me 3x squared. X times 12 will give me 12x.

So now that I have a common factor in both the numerator and the denominator, I can see that they will cancel each other out. And so this will simplify to be 8x squared minus x over 3x plus 12. And I'll check. My terms do not all have a common factor, so this is as simplified as my answer can be.

So finally let's do an example dividing rational expressions. I've got x plus 4 over x to the third divided by 5x plus 1 over 2x minus 1. If you're feeling confident, go ahead and pause. And then check back later to see if you got the right answer.

So when we're dividing rational expressions, just as we divide numerical fractions we're going to multiply by the reciprocal. So this is going to become x plus 4 over x to the third multiplied by 2x minus 1 over 5x plus 1. So I simply reverse the numerator and the denominator to find the reciprocal.

So now that this is a multiplication problem, I can simply multiply my numerators together and my denominators together. In my numerators, I have two binomials, so I'm going to use FOIL to multiply. x times 2x will give me 2x squared.

x times negative 1 will give me minus 1x, or minus x. 4 times 2x will give me 8x. And 4 times negative 1 will give me negative 4.

In my denominator, I'll just need to distribute, multiply x to the third by both terms in my second fraction. So x to the third times 5x will give me 5x to the fourth power. And x to the third times 1 give me plus 1x to the third, or just x to the third.

I can simplify my numerator by combining these two terms. Negative 1x plus 8x is going to give me 7x minus 4. And now checking for any common factors amongst all of my terms, I see that I don't have any common factors, so this expression is as simplified as my answer can be.

So let's go over our key points from today. A rational expression is a fraction in which the numerator and denominator are polynomials.

To multiply rational expressions, multiply the numerators of fractions together and the denominators of the fractions together. This is the same process as multiplying numerical fractions. To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. This is the same process as dividing numerical fractions.

So I hope that these key points and examples helped you understand a little bit more about multiplying and dividing rational expressions. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.