Question Video: Division of Rational Expressions Mathematics

Video Transcript

Simplify five 𝑥 squared minus seven 𝑥 over three 𝑥 minus two divided by two 𝑥 minus five over four 𝑥 minus seven. Write your answer in fully factored form.

So the first thing that we’ll need to do is actually apply one of our fraction operation rules because actually in the question you can see that we’ve got an algebraic fraction divided by another algebraic fraction. And actually, it doesn’t matter if there are algebraic fractions. We still apply the same rules. So if we got 𝑎 over 𝑏 divided by 𝑐 over 𝑑, then this is equal to 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐.

So therefore, we will rewrite our expression. What we have is five 𝑥 squared minus seven 𝑥 over three 𝑥 minus two and now multiplied by four 𝑥 minus seven over two 𝑥 minus five which is actually the reciprocal of two 𝑥 minus five over four 𝑥 minus seven. Okay, great, so now, what do we do next?

So again, we’re actually gonna apply one of our original fraction operation rules. And this one is if we’re actually multiplying fractions, then all we do is we multiply the numerators and denominators. So for instance, if we have 𝑎 over 𝑏 multiplied by 𝑐 over 𝑑, then it’s just equal to 𝑎𝑐 over 𝑏𝑑. So therefore, if we apply this, our expression is gonna be equal to five 𝑥 squared minus seven 𝑥 multiplied by four 𝑥 minus seven over three 𝑥 minus two multiplied by two 𝑥 minus five.

Okay, fab, so we’ve actually simplified. But have we actually finished the question? Have we done everything that we need to do? Well, this is when I always say “let’s take a look back at the question and see what it wanted.” But it actually says that it wants the answer in fully factored form. So therefore, we can actually see that our first parenthesis is actually not fully factored. So we can actually factor this to actually help us reach our final answer.

Well, if we take a look at five 𝑥 squared minus seven 𝑥, we can see that 𝑥 is actually in fact a factor of both terms because we have five 𝑥 squared — so there is an 𝑥 there — and negative seven 𝑥 — so again we have an 𝑥. So therefore, we’ll take 𝑥 outside the parentheses. So we have 𝑥 multiplied by five 𝑥 because five 𝑥 multiplied by 𝑥 gives us our five 𝑥 squared and then negative seven because negative seven multiplied by 𝑥 gives us our negative seven 𝑥.

Okay, great, so let’s use this to actually write out our fully simplified expression. So therefore, we can say that if we simplify five 𝑥 squared minus seven 𝑥 over three 𝑥 minus two divided by two 𝑥 minus five over four 𝑥 minus seven and write it in fully factored form, the answer is gonna be equal to 𝑥 multiplied by five 𝑥 minus seven multiplied by four 𝑥 minus seven over three 𝑥 minus two multiplied by two 𝑥 minus five.