- In previous math courses, we learned how to do long division with numbers, including recognizing quotients and remainders. How do we do this with polynomials?
- Is there a relation between the degrees of the polynomials involved in the division and the degrees of the quotient and remainder?
Introduction
Let’s recall some terminology from division of numbers. If we divide by , we have that the quotient is and the remainder is . In other words, , which can be also written as In general, when and are positive integers, performing the division of by gives us some quotient and some remainder , where the remainder is less than . We can express this as . This sort of idea also works if, instead of numbers, we consider polynomials. If and are polynomials, we’ll understand how to find polynomials and such that with the degree of less than the degree of , and the degree of equal to the difference between the degrees of and . Note that if is a point where , then we can also write this relationship as This is extremely useful when trying to study rational functions and their asymptotes, since these functions are by definition quotients of polynomials.
Long division of polynomials
The best way to understand this is through guided examples.
- First, what do we need to multiply to in order to get the leading term of ? We can see that works, since , so we know that .
- In the diagram, we place above the term, then underneath subtract by :
- Now we repeat this with the polynomial on the bottom (namely ) in place of . So, what do we need to multiply to in order to get the leading term of ? This time, we need just , so we know that .
- We now add to the polynomial at the top and subtract from the polynomial at the bottom:
- Now the polynomial at the bottom is , which is degree 0. Since the degree is strictly less than the degree of , we’re done.
- Our quotient is the polynomial at the top, , and our remainder is the polynomial at the bottom, . Therefore we may write or equivalently,
- What do we need to multiply to in order to get the leading term of ? Since and , we should multiply by . So, we know that .
- In the diagram, we write above and subtract from to get
- Now we repeat with the bottom polynomial, , in place of . What do we need to multiply to in order to get the leading term of ? This time, will work, so .
- In the diagram, add to the polynomial at the top, and subtract the bottom by to get
- Since the polynomial at the bottom is and thus has degree 1, we know we’re done since this is strictly less than the degree of . Therefore our quotient is and our remainder is . Written out fully, we have or equivalently,
More on Remainders
- is a factor of ,
- divides evenly into , or
- the remainder of is zero.
Notice that each of these points are different ways of saying the same thing.
This can be extended to values which are not zeros of as well. Namely, if is any real number, then is equal to the remainder of .
- Long division of polynomials works essentially like long division of numbers.
- When performing the long division of by and writing the relation , we know that the degrees of and add to the degree of , and the degree of is strictly less than the degree of (this tells us when to stop dividing).
- We can detect roots of a polynomial by dividing by the polynomial with no remainder, and conversely, we can detect that is a factor of if is a root.