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Video: Dividing Two Rational Expressions

Kathryn Kingham

Answer the following questions for the rational expressions (5𝑥³ − 45𝑥)/(12𝑥² − 4𝑥) and (15𝑥 − 45)/3𝑥². Evaluate (5𝑥³ − 45𝑥)/(12𝑥² − 4𝑥) divided by (15𝑥 − 45)/3𝑥². Is the result a rational expression? Would this be true for any rational expression divided by another rational expression?

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Video Transcript

Answer the following questions for the rational expressions five 𝑥 cubed minus 45𝑥 over 12𝑥 squared minus four 𝑥 and 15𝑥 minus 45 over three 𝑥 squared.

First we need to evaluate five 𝑥 cubed minus 45𝑥 over 12𝑥 squared minus four 𝑥 divided by 15𝑥 minus 45 over three 𝑥 squared, then we need to answer the question is this result a rational expression and the question would this be true for any rational expression divided by another rational expression.

Before we get started, we’re going to look at each of these fractions separately and see if they can be reduced before we divide them. For example, in the numerator here, there’s a factor of five in each of the terms, in five 𝑥 cubed and in 45𝑥. There is also a common factor of 𝑥 in the numerator. By taking out the common factor of five 𝑥, we can simplify the numerator to say five 𝑥 times 𝑥 squared minus nine.

The denominator has a common factor of four and also a common factor of 𝑥. If we take out the factor four 𝑥, we’ll be left with three 𝑥 minus one. From here, we see that our numerator and our denominator have an 𝑥 in it. 𝑥 over 𝑥 equals one; those 𝑥s cancel each other out.

And then we notice one more thing: 𝑥 squared minus nine is the difference of squares; 𝑥 squared is a square, and nine is a perfect square. This means in our numerator we can change 𝑥 squared minus nine to 𝑥 squared plus three times 𝑥 squared minus three. There’s nothing else we can simplify in our denominator, so we just bring that down.

Now we’ve simplified the first rational expression. We wanna do the same thing for our next rational expression. Here we have both 15 and 45 being divisible by 15. If we take out that common factor of 15, we’ll be left with 15 times 𝑥 minus three. Our denominator is already in its most simple form, so we just bring that down. Then I notice that 15 divided by three equals five, so I cancel out that 15 over three, and I’m left with five over one.

And now our second rational expression is completely reduced. To divide those two fractions, we follow the rules we always use when were dividing fractions. And that rule says to keep the first term the same, and then change the division to multiplication, multiply, by the reciprocal. Now I can see I have a five in the numerator and in the denominator; those will cancel out.

I also have an 𝑥 minus three in the numerator and an 𝑥 minus three in the denominator; those will cancel out. From here, I follow the rules of multiplying fractions and that means I multiply the numerator by the numerator and the denominator by the denominator. Each one of my numerators only has one term, so I multiply 𝑥 squared by 𝑥 plus three.

And the denominator will be four times three 𝑥 minus one times one for our final answer of 𝑥 squared times 𝑥 plus three over four times three 𝑥 minus one. Is the result a rational expression? Yes, a rational expression is an expression that has a polynomial in the numerator and/or the denominator, and this fits that description.

Would this be true for any rational expression divided by another rational expression? What do you think? Yes! Any time we multiply or divide a rational expression by another rational expression, it’s going to result in a rational expression.