Polynomial Division

Polynomial division is a fundamental operation in algebra and is used to simplify or factor polynomials. There are two common methods for dividing polynomials: long division and synthetic division. In this lesson, we will cover both methods and show you how to divide polynomials.

1. Long Division of Polynomials:

Long division for polynomials is similar to long division for numbers. Here are the steps:

Step 1: Arrange the polynomial terms in descending order of degrees.

Step 2: Divide the first term of the dividend (numerator) by the first term of the divisor (denominator) to determine the leading term of the quotient.

Step 3: Multiply the divisor by the leading term of the quotient and subtract the result from the dividend. Write the remainder as a fraction over the divisor.

Step 4: Bring down the next term from the dividend and repeat the process until you have dealt with all terms in the dividend.

Step 5: The final result is the quotient, and any remainder is written as a fraction over the divisor.

2. Synthetic Division:

Synthetic division is a more concise method for polynomial division when dividing by linear factors (polynomials of the form xc).

Step 1: Write the coefficients of the dividend polynomial (including any missing terms with zero coefficients) and the divisor as

(xc), where c is the constant you would set equal to zero to find the root.

Step 2: Determine the value of c. In the divisor (xc), set (xc)=0, and solve for c.

Step 3: Perform synthetic division by bringing down the first coefficient, multiplying it by c, and adding the result to the next coefficient. Continue this process until you reach the last coefficient.

Step 4: The results of synthetic division provide the coefficients of the quotient (excluding the remainder). The last value in the row is the remainder.

            x^2 - x + 1


x - 4  |  x^3 - 5x^2 - 4x + 20

            - (x^3 - 4x^2)


                     - x^2 - 4x

                      + (x^2 - 4x)



The quotient is

x^2−x+1, and there is no remainder. 

Synthetic Division:

Set c=4, and write the coefficients of the dividend polynomial:

4 | 1  -5  -4  20

Perform synthetic division: 

   1  -1  -5 | 20

     | 4   12 | -20

The quotient coefficients are 1, -1, and -5, and the remainder is -20.

Both long division and synthetic division can be useful in different scenarios, so it's important to understand and be proficient in both methods.