The division of two polynomials is similar to the division of two integer numbers: as a result of the division, we get another polynomial and a remainder whose degree is less than that of the divisor.
Remember that, given a field (e.g., the set of real numbers or the set of complex numbers ) and a non-negative integer , a polynomial of degree is a function defined bywhere both the argument and the coefficients belong to and .
We denote the degree of a polynomial by .
In this section and the next one we are going to revise some facts about sums and products of polynomials.
Recall that two polynomials are summed by summing their coefficients.
If is the polynomial of degree defined above and is another polynomial of degree defined bythen
The degree of is less than or equal to the degree of the higher-degree polynomial . In particular, note that if and , then the degree of is less than .
Example Let be the polynomial of degree defined by and the polynomial of degree defined byThen, the sum is the polynomial of degree defined by
As before, letThe product iswhere the coefficients satisfy
Since and (otherwise and would not be polynomials of degree and respectively), and is a polynomial of degree .
Example Let and be the polynomials defined in the previous example. Then,and
The following proposition goes under the name of Division Algorithm because its proof is a constructive proof in which we propose an algorithm for actually performing the division of two polynomials.
Proposition Let and be two polynomials and . Then, there exists a unique polynomial such that and . Moreover, when , or when .
Letandwhere and . If , then we obtain a representation of the desired form by setting and . If , then we definewhere because . By multiplying the polynomial by , we transform it into a polynomial having the same degree as and the same leading coefficient . Thus, has at least one degree less than . We now havewhereSince , if , we are done. Otherwise, we definewhere is the leading coefficient of and . We now havewhere we have definedNote that Therefore, if , we are done. Otherwise, we keep lowering the degree of the remainder with the same technique used in the previous steps, until in step we obtainwhere has degree and . We now prove uniqueness by contradiction. Suppose that there are two representations having the desired characteristics:where . Then, we haveSince , the polynomial has degree greater than or equal to . The same must be true of or . But this is impossible because the degrees of and must be less than . Hence we have a contradiction, and it must be that .
We use the following terminology for the four polynomials involved in the division algorithm
is the dividend;
is the divisor;
is the quotient;
is the remainder.
If the remainder obtained at the end of the division algorithm is identically zero, that is, andthen we say that divides , that is a divisor of or that is divisible by , and we write
Note that the requirement that means that the zero polynomial can never be a divisor.
Note that if the dividend is identically equal to zero, then, for every with , we have
Therefore, for every
Let us make an example to see how the division algorithm works in practice.
In order to understand the example, we need to first read the proof of the Division Algorithm (above), where the steps of the algorithm are explained.
Example Let us divide the polynomialby the polynomialHere and . We define Thus,andThe degree of is larger than that of . As a consequence, we need to further reduce the degree of the remainder. The previous calculations are displayed synthetically as where we have reported the dividend in the second row, the divisor in the leftmost column, the quotient in the first row, the product in the third row and the remainder in the last row. The next steps of the algorithm are performed directly on the tableau:In the last row, the degree of the remainder is less than that of the divisor, so we can stop the algorithm. The result of the division is
Here is a simple but extremely useful fact about polynomials that are divisors of each other.
Proposition Let and be two polynomials not identically equal to zero. If and , then where is a non-negative constant belonging to .
The fact that implies that there is no remainder in their division. By looking at the Division Algorithm, we can see that this happens only in the case in which (it also happens when and , but by assumption). Similarly, the fact that implies that . Therefore, we have thatLetwhere is the quotient of the division. By the properties of the multiplication of polynomials (reviewed above), we haveTherefore,Thus, is a polynomial of degree zero, that is, a non-negative constant.
Here is an application of the Division Algorithm, called Remainder Theorem.
Proposition Let be a polynomial and . If we divide by , we obtain as a remainder.
By applying the Division Algorithm, we obtainwhere the degree of needs to be lower than the degree of . The degree of the latter is . Therefore, is a constant. Thus, we haveor
Here is a corollary of the previous proposition, known as Factor Theorem.
Proposition Let be a polynomial and . Then, is divisible by if and only if is a root of .
By definition, is divisible by if and only if the remainder of the division is zero, which, by the Remainder theorem, happens if and and only if , which in turn happens if and only if is a root of .
Below you can find some exercises with explained solutions.
Divide the polynomialby the polynomial
We need to perform only a single step: The result of the division is
Please cite as:
Taboga, Marco (2017). "Division of polynomials", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/division-of-polynomials.
Most of the learning materials found on this website are now available in a traditional textbook format.