How to Add and Subtract Rational Expressions — College Algebra Tips

How to Add and Subtract Rational Expressions — College Algebra Tips


In this exercise we’re asked to subtract
the rational expressions 3 over x minus six minus x plus 15 over x minus 6 times
x plus 1. Remember that rational expressions are just big fractions, so
just like with adding and subtracting fractions, you need a lowest common
denominator. Well in order to get a common denominator, the first step with
rational expressions is to factor each of the denominators. It’s already done
for us here – you have x minus 6 times the quantity x plus 1 in the second fraction,
or rational expression. And this first one is x minus 6 – you can’t factor that
anymore to get a common denominator. Think about building up the denominators;
that is what is this missing what factor is this missing that this one has? Well
it’s x plus 1 so we’re going to multiply both the numerator and the denominator
by x plus 1 just like you would multiply the numerator and the denominator by
something when you were adding or subtracting fractions. Now we’re going to
multiply the 3 times the x plus 1 because we are multiplying the numerator
and the denominator by x plus 1. So distribute that three and what we’re
going to get is a 3x plus 3 in the numerator all over the denominator of x
plus 1 times x minus 6. And now, because we have a common denominator, I can just
continue this because we know when we add or subtract fractions you have a
common denominator and then you add or subtract straight across the numerator.
So I’m just going to continue this minus because of this minus sign. Now remember
we are subtracting that entire numerator so that’s why I put that in parenthesis – so minus the quantity x + 15. Here let’s think about this minus sign;. You’re going
to distribute that – you can give it as a negative one if you’d like
to, whatever helps you – but this is going to change the sign of everything inside
the parentheses so the result now is going to be 3x plus 3 minus x minus 15
all over that common denominator of x plus 1 times x minus 6. And by the way,
it’s okay to write x plus 1 before x minus 6. Just like 8 times 5 is equal to
5 times 8, this is the commutative property of multiplication – you can write
these in the opposite order. Now let’s combine some like terms. So 3x minus x
gives us 2x and positive 3 minus 15 gives you negative 12 all over that
common denominator of x plus 1 times x minus 6. Notice that i did not multiply
those factors in the denominator and, I’ll show you why right now, when you
simplify a rational expression you want to factor both the numerator and the
denominator. So that’s why we left that denominator factored because we were
going to end up simplifying that anyway. And now let’s factor the numerator and
that is 2, is the common factor that we can factor out, times x minus 6, is the
quantity that’s left. So 2 times the quantity x minus 6 equals 2x minus 12 over
that denominator x plus 1 times the quantity x minus 6. Hey surprise surprise –
we have a factor that is common that we can cancel out of the numerator and the
denominator, and look out for that – that’s probably going to happen a lot in
problems. They’re kind of set up so that you can cancel something out – not always,
but look for it. So we have 2 over x plus 1 as our final answer.

One thought on “How to Add and Subtract Rational Expressions — College Algebra Tips

  • February 29, 2012 at 6:58 pm
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    If you asked me if I knew how to do this, I would have said I'd forgotten this decades ago. But it turns out, with a little assist from the narrator, I actually remember. Cool.

    Reply

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