Question Video: Dividing Two Rational Expressions

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. Evaluate five 𝑥 cubed minus 45𝑥 over 12𝑥 squared minus four 𝑥 divided by 15𝑥 minus 45 over three 𝑥 squared. Is the result of five 𝑥 cubed minus 45𝑥 over 12𝑥 squared minus four 𝑥 divided by 15𝑥 minus 45 over three 𝑥 squared a rational expression? Would this be true for any rational expression divided by another rational expression?

We have two rational expressions, and we want to divide one by the other. To divide this rational expression by another rational expression, we need to multiply by the reciprocal. This leaves us with five 𝑥 cubed minus 45𝑥 over 12𝑥 squared minus four 𝑥 times three 𝑥 squared over 15𝑥 minus 45.

At this point, we could do some simplifying. The numerator of our first rational expression we can rewrite as five 𝑥 times 𝑥 squared minus nine. But as we notice that we have a difference of squares, we can further expand 𝑥 squared minus nine into the factors 𝑥 plus three times 𝑥 minus three. And in the denominator of our first rational expression, we can take out a factor of four 𝑥, leaving us with four 𝑥 times three 𝑥 minus one. The numerator of the second expression remains three 𝑥 squared. We can rewrite the denominator as 15 times 𝑥 minus three. We find an 𝑥 minus three term in the numerator and the denominator. We have an 𝑥-term in the numerator and the denominator.

And then notice that we have a 15 in the denominator. In the numerator, we have five times three. As five times three is 15, 15 over 15 equals one, which means in the numerator we now have 𝑥 plus three times 𝑥 squared. We can write this as 𝑥 squared times 𝑥 plus three. And in the denominator, we have four times three 𝑥 minus one. This completes our step one. When we divide these two expressions, we get 𝑥 squared times 𝑥 plus three over four times three 𝑥 minus one.

For the second part of the question, we need to know if this result is also itself a rational expression. The definition of a rational expression is an expression that has a polynomial in the numerator and the denominator. Here we have a polynomial in the numerator and a polynomial in the denominator, which means yes, our result is a rational expression.

Would this be true for any rational expression divided by another rational expression?

To answer this, let’s consider a counterexample. Let’s see if we can think of any case where a rational expression divided by another rational expression would not be itself a rational expression. Let’s use a constant five-fourths as our first rational expression. And our second rational expression is zero over 𝑓 of 𝑥, zero over some function of 𝑥. When we try to divide by zero over 𝑓 of 𝑥, we multiply by 𝑓 of 𝑥 over zero. And we end up with a zero in the denominator, which is undefined and is not a rational expression. By finding one counterexample, we can say that this statement would not be true for any rational expression divided by another rational expression.

To summarize, we took two rational expressions, divided one by the other. The result was itself a rational expression. However, we saw that this would not always be true.