Use polynomial division to simplify
six 𝑥 cubed plus five 𝑥 squared minus 20𝑥 minus 21 over two 𝑥 plus three.
One method we do have for
simplifying algebraic fractions is to factor where necessary. It’s not particularly
straightforward to factor this cubic on our numerator. So instead, we’re going to recall
that this line in a fraction actually just means divide. And we’re going to use polynomial
long division. Our dividend, that’s the numerator
of our fraction, goes inside the bus stop. The divisor, that’s the
denominator, goes on the outside.
And then, we remember the first
thing that we do is we take the first term in our dividend, that’s six 𝑥 cubed, and
we divide it by the first term in our divisor, that’s two 𝑥. Six divided by two is three. Then, if we consider 𝑥 as being 𝑥
to the power of one, we know that we can subtract these exponents. And 𝑥 cubed divided by 𝑥 to the
power of one is 𝑥 squared. This means that six 𝑥 cubed
divided by two 𝑥 must be three 𝑥 squared.
Our next step is to multiply three
𝑥 squared by each term in our divisor. Three 𝑥 squared times two 𝑥 is
six 𝑥 cubed. Notice that this is the same as the
first term in our dividend, so we know we’ve probably started this correctly. We then calculate three times three
𝑥 squared. Well, that’s nine 𝑥 squared. Our next step is to subtract six 𝑥
cubed plus nine 𝑥 squared from six 𝑥 cubed plus five 𝑥 squared. Six 𝑥 cubed minus six 𝑥 cubed is
zero. We don’t really need to write this
zero. And then, five 𝑥 squared minus
nine 𝑥 squared is negative four 𝑥 squared. We bring down the next term. Some people bring down all of the
terms, but I prefer to keep things a little bit simpler.
And we’re now going to divide
negative four 𝑥 squared by two 𝑥. Negative four divided by two is
negative two. 𝑥 squared divided by 𝑥 to the
power of one is 𝑥. So, negative four 𝑥 squared
divided by two 𝑥 is negative two 𝑥. And we add negative two 𝑥 above
five 𝑥 squared in our problem. Now, we multiply negative two 𝑥 by
each term in our divisor. Two 𝑥 multiplied by negative two
𝑥 is negative four 𝑥 squared, and negative two 𝑥 times three is negative six
We then subtract each of these
terms from negative four 𝑥 squared minus 20𝑥. Negative four 𝑥 squared minus
negative four 𝑥 squared is negative four 𝑥 plus four 𝑥 squared. So that’s zero, and we don’t really
need to write that. We then do negative 20𝑥 minus
negative six 𝑥. That’s negative 20𝑥 plus six 𝑥,
which is negative 14𝑥. We bring down negative 21. And we’re now going to divide
negative 14𝑥 by two 𝑥. 𝑥 divided by 𝑥 is just one. So, we get negative 14 divided by
two, which is negative seven. So, we add negative seven here. And once again, we divide this
number by each term in our divisor. Negative seven times two 𝑥 is
negative 14𝑥, and negative seven times three is negative 21.
We do one final subtraction, and
this is a really important step to do because it tells us whether there’s a
remainder or not. In fact, negative 14𝑥 minus 21
minus itself is just zero. And so, we’ve completed the
division. When we simplify our algebraic
fraction, we’re left with three 𝑥 squared minus two 𝑥 minus seven.
Now, at this stage, it’s really
useful just to discuss briefly how we might check our solution. We perform an inverse
operation. We take our quotient, here that’s
the solution to the division, and multiply that by the divisor, remembering, of
course, that the divisor is the algebraic expression here that we’re dividing
by. We multiply three 𝑥 squared minus
two 𝑥 minus seven by two 𝑥 plus three. And when we do, we should get the
numerator, or the dividend.