Polynomials division calculator

Consider the long division method in the following example. Suppose we need to divide the polynomial

by a polynomial

At first, it should be noted that:

division of polynomials is possible only if the degree of the divisible polynomial is greater than or equal to the degree of the divisor polynomial.

In our case, this condition is satisfied because the degree of the divisible polynomial is three, and the degree of the divisor polynomial is two.

To divide the polynomials, write the divisible polynomial to the left of the vertical line, and the divisor polynomial to the right:

Next, we divide the term with the highest degree of the dividend polynomial by the term with the highest degree of the divisor polynomial :

Write the result (the quotient of division) to the right under the horizontal line:

Now, multiplying by the divisor polynomial , we get:

Write the obtained result on the left, under the dividend polynomial:

Subtract the result from the dividend polynomial:

Write the resulting polynomial on the left under the line:

Next, the procedure is repeated, i.e. we divide the term with the highest degree of the resulting polynomial ( ) by the term with the highest degree of the divisor polynomial ( ), etc., as a result we get:

The division process stops when the degree of the remainder polynomial is less than the degree of the divisor polynomial. This condition is described above.

We write the result as follows. First, we write the quotient (the polynomial on the right under the line) equal to , then we add to it a fraction whose numerator is the remainder polynomial equal to (the one that remains after all the subtractions on the bottom left in the column) and the denominator is the divisor polynomial . As a result, we get:

Thus:

Another way to divide polynomials is the method of undetermined coefficients. Consider the same example. In general, the result of the division of the polynomials can be written in the following form:

where is a quotient polynomial whose degree is equal to the difference between the degrees of the dividend polynomial and the divisor polynomial, i.e., in our case, the degree is one. is a remainder polynomial whose degree is not greater than the degree of the divisor polynomial, i.e., in our case, it is not greater than one.

Now, write the polynomial in general form:

- undetermined coefficients so far.

The same is true for the polynomial :

- undetermined coefficients so far.

Thus, we obtain the following equality:

So, we need to determine the unknown coefficients and . To do this, multiply both parts of the above equation by the denominator (the divisor polynomial) , we get:

Open the brackets, find similar terms:

In order to maintain the correct equality, we need to equate the coefficients at the same degrees . As a result, we get the following system of linear equations:

After solving this system, we get the following values of the coefficients:

Substitute the values of the obtained coefficients and the initial equality:

As you can see, this result completely coincides with the result obtained by the long division method.