Free Homework Math Help: Polynomial Division

Polynomial Division

There are basically two ways to divide a polynomial by another polynomial. The first method is long division. It parallels the rugular division method in arithmetic. When one number is divided by another, there is a quotient and a remainder. The form can be written as:

p d

=   q   +

r d

or   p= dq + r

For polynomials, it is written as:

P(x) D(x)

=   Q(x)   +

R(x) D(x)

or

P(x) = Q(x)D(x) + R(x)

where

P(x), Q(x), D(x) and R(x) are functions.

Long Division

Polynomial Division Method

1. Put the smaller degree polynomial to the left and the larger degree polynomial underneath the division line.
2. Divide the first term of the large polynomial by the first term of the small one. Write the answer avove the line and above the term being divided.
3. Multiply the term resulting from the previous division times the smaller polynomial.
4. Subtract the result of the multiplication from the large polynomial.
5. Divide the first term of the small polynomial into the remaining part from the subtraction.
6. Repeat steps 3, 4 and 5 until either no remainder results or until it is smaller than the small polynomial.

			Divide the leading term of the divisor into the leading term of the dividend and write the result above the leading term of the dividend.
			Multiply the result with the divisor and write the product below the dividend with the terms of the same degree lined up vertically.
			Subtract the product from the divisor by multiplying the product times -1 and then adding.
			Bring down the next term from the dividend.
			Divide the leading term of the divisor into the leading term of the remainder and write the result above. Repeat the multiplication of the result with the divisor.
			Repeat the subtraction process.
			When the result of subtraction is a constant, it becomes part of the fraction for the final remainder.
			The constant becomes the numerator while the divisor goes in the denominator for the fraction part of the answer.
			Check the answer by multiplying the resulting answer with the divisor to see if it equals the dividend.

Synthetic Division

Synthetic division is similar to long division except there are no variables, just coefficients. The coefficients of the polynomial are written down. The number being divided into the polynomial is written to the left of the coefficients and is separated by a line. Another line separates the coefficients along with a blank row from the resulting answers. The leading coefficient is brought down and multiplied by the number on the left. The result is placed in the blank row under the next coefficient. These are added and the result of the addition is carried below the line. The process is continued until the final remainder is left. The remainder can determine if the answer is a root of a polynomial.

Divide the polynomial 2x⁵ + x⁴ + 3x² + 1 with x + 2.

			The first step in the synthetic division process is to drop down the leading coefficient. Also, notice that zeros are used as placeholders for the missing terms in the dividend.
			The next step is to multiply the dropped coefficient with the number from the divisor and place the result above the line below the next leading coefficient.
			Next, add the product with the next leading coefficient.
			Multiply the result of the addition with the number from the divisor
			Continue the process until the last result. This is the remainder.

Another Synthetic Division Example

Synthetic division can be used to test factors of a polynomial. The cubic polynomial x³ + 3x² + 3x + 1 has ( x + 1 ) as a factor. If a number is a factor then the remainder after synthetic division with that factor will be zero.

The remainder of synthetic division is zero. A remainder of zero indicates that -1 is a root of the polynomial.