Polynomial division is the process whereby a polynomial, , is divided by a polynomial, , usually of degree less than or equal to that of . The result is of the form
where is the quotient polynomial and the remainder polynomial of degree less than the degree of .
If then is said to divide exactly.
There are two common methods of polynomial division: long division and factorisation.
The process is very like long division of numbers, except with algebra. This method is best explained using an example.
Divide by .
Begin by writing the question out in long division form:
Starting with the first term in the polynomial, , calculate what needs to be multiplied by so that when we subtract it from the polynomial, it will eliminate the term. In this case it is , as , so we subtract this from the polynomial under the division sign. We write the on top of the division line.
We now have a new polynomial with which to repeat the process.
This time we are looking to eliminate , so we multiply by to get which we can then subtract to get rid of the term.
Finally, we look to eliminate so we multiply our divisor by to get , and subtract.
This happens to have eliminated the constant as well and we are left with zero. This means that is a factor of the cubic. We have shown
Divide by .
Begin by writing the question out in long division form.
Starting with the first term in the polynomial, , calculate what needs to be multiplied by so that when we subtract it from the polynomial, it will eliminate the term. In this case it is , as , so we subtract this from the polynomial under the division sign. We write the on top of the division line.
We now have a new polynomial with which to repeat the process.
This time we are looking to eliminate , so we multiply by to get which we can then subtract to get rid of the term.
Now, we look to eliminate so we multiply our divisor by to get , and subtract.
We cannot divide this any more and are left with a remainder of . So
Prof. Robin Johnson uses the long division method to divide by .
In this example, Prof Johnson finds all the roots of a cubic equation, with one of the roots already known, by first finding a factor of the cubic given by the known root and then dividing by that factor to obtain a quadratic and then find its roots to find the remaining two roots of the cubic.
The factorisation method involves forcing each term in to have a factor .
Consider the case when is a polynomial of degree one, i.e. of the form . For each term in , replace it with . This gives us one term with as a factor, and a second term of lower degree which can be combined with the other term in the original expression. After applying this method to each non-constant term, we end up with an expression whose terms all have a factor of , apart from the constant term (if there is one).
Divide by .
First, we want to take a factor of out of the term and replace it with . , so we need to subtract an term in order to have the same quantity as we started with. We now have
Now we consider the term. As before, we take out a factor of and replace it with . , so we need to add to preserve equality. We now have
Now every term in the expression has a factor of so it's straightforward to divide by .
Note: There is no remainder, so we have shown that is a factor of .
Divide by .
First, consider the term . Take out a factor of and replace it with . , so we must add to preserve equality.
Now consider the term. This is replaced with .
Finally, consider the term. This is replaced with .
Now as many terms as possible have a factor of , so it's straightforward to perform the division:
Prof. Robin Johnson uses the factorisation method to divide by , and by .
In this example, Prof Johnson finds all the roots of the cubic equation with one of the roots already known, by first finding a factor of the cubic given by the known root and then dividing by that factor to obtain a quadratic. Finally, he solves the quadratic to give the remaining two roots of the cubic.
Test yourself: Numbas test on polynomial division