Video Transcript
Simplify 𝑥 plus four over 𝑥 minus
one multiplied by 𝑥 plus one over 𝑥 squared plus three 𝑥 minus four.
So, whenever we’re trying to solve
this type of problem, before we look at multiplying numerators and multiplying
denominators, what we want to do is see if any of the terms that we’ve got can be
factored. Well, in fact, we can see yes,
there’s one that can be because the denominator from the right-hand side is a
quadratic. Well, we think it can be
factored. But let’s check. Because even though it’s a
quadratic, this doesn’t automatically mean it can be factored. So, we’ve got 𝑥 squared plus three
𝑥 minus four.
Now, to remind ourselves what we do
if we want to factor quadratic, what we’re looking for is two numbers whose product
would give us, in this case, the negative four and whose sum would give us positive
three. Well, the two values we need would
be four and negative one. That’s because four multiplied by
negative one is negative four and four add negative one is positive three. So therefore, we know that this can
be factored. So, 𝑥 squared plus three 𝑥 minus
four becomes 𝑥 plus four multiplied by 𝑥 minus one.
So now, what we can do is
substitute this back into our expression. So when we do that, what we have is
𝑥 plus four over 𝑥 minus one multiplied by 𝑥 plus one over 𝑥 plus four
multiplied by 𝑥 minus one. Now the reason why we do this while
we’re factoring a thing that can be factored is to search for any common factors on
the numerators and denominators. And now, once we’ve done this, we
can see that we’ve got one because we’ve got 𝑥 plus four in the numerator of the
left-hand side and 𝑥 plus four in the denominator of the right-hand term. So now, what we can do is divide
through by this factor, which we’ve done. So now, what we’ve got is one over
𝑥 minus one multiplied by 𝑥 plus one over one multiplied by 𝑥 minus one.
So, now, what we need to do is
multiply the numerators and multiply the denominators. So, when we do that, we’re left
with our fully simplified answer, which is 𝑥 plus one over 𝑥 minus one
squared. And we got 𝑥 minus one squared in
the denominator because we had 𝑥 minus one multiplied by 𝑥 minus one.
