Answer the following questions for the rational expressions six times 𝑥 minus two over three 𝑥 squared minus six [𝑥] and six 𝑥 minus three over two 𝑥. The questions are find the product of six times 𝑥 minus two divided by three 𝑥 squared minus six 𝑥 and six 𝑥 minus three over two [𝑥]. Then we need to know is the product a rational expression and would this be true for the product of any two rational expressions.
The first step tells us to multiply these rational expressions. So let’s start there. Before we multiply these rational expressions, let’s see if we can reduce either one of the terms. I notice that in the denominator here both three 𝑥 squared and six 𝑥 have a common factor of three; they also have a common factor of 𝑥.
If we pull out that common factor of three 𝑥, we can rewrite the denominator in an equivalent form of three 𝑥 times 𝑥 minus two. We just bring down the numerator. We haven’t changed the numerator. And all of a sudden, we can see that the numerator and the denominator have a factor of 𝑥 minus two. 𝑥 minus two divided by 𝑥 minus two would be equal to one; those factors cancel out.
I also notice that six over three can be reduced. Six over three can be reduced to two over one. Our first rational expression can be reduced to two over 𝑥. And the numerator of our next rational expression I notice a common factor of three. If we take out the factor of three, we can rewrite the expression three times two 𝑥 minus one over two 𝑥. It doesn’t look like there’s anything that can be cancelled out in this expression. So we can just bring it down to the next line. As I did that, I recognise that there’s a two in the numerator of one expression and in the denominator of another expression. Two over two would be equal to one. So those two cancel out.
Now we can multiply. Remember that when we’re multiplying fractions, we multiply their numerators by each other and their denominators by each other. One times three 𝑥 times two 𝑥 minus one looks like this and 𝑥 times 𝑥 would be 𝑥 squared. The product for our first part of this question equals three times two 𝑥 minus one over 𝑥 squared.
The next question wants to know is the product a rational expression. A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Our product has a polynomial in the numerator, which makes it a rational expression.
The last question is asking us this would this be true for the product of any two rational expressions. The question is asking us if we multiply two fractions with polynomials in the numerator and/or the denominator, would they produce a fraction with polynomials in the numerator and/or the denominator. Yes, this would be true for any two rational expressions.
Just a note that’s not really part of this question, but it’s important to remember that for this expression 𝑥 cannot be equal to zero because we can’t divide by zero. So anytime you see an 𝑥 or an 𝑥 squared in the denominator in an expression like this, we always know that 𝑥 cannot be equal to zero. There’s not a solution at that point.