How to Divide Polynomials Using Synthetic Division

Synthetic division is a shortcut method of dividing a polynomial by a linear factor, but it's real purpose is in finding the zeros of a polynomial. It requires some guesswork, but in many cases it's faster and easier to find the zeros with synthetic division than by working out the factors of the polynomial.

Instructions

1. Estimate the possible zeros. Good guesses are the factors of the coefficients of the polynomials.

2. Write the coefficients of the polynomial inside an upside-down division sign. Leave some room between the coefficients and the horizontal line for another row of numbers.

3. Place one of the possible zeros to the left of the vertical line.

4. Copy the first coefficient under the horizontal line. Multiply this number by the zero and write it on top of the horizontal line, under the second coefficient.

5. Add this number to the second coefficient and write the sum under the horizontal line. Multiply this number by the possible zero and place the result above the horizontal line, under the third coefficient.

6. Add this number to the third coefficient and write the sum under the horizontal line. If the sum is zero, then you have found a zero of the polynomial, and the numbers under the horizontal line are the coefficients of a factor of the polynomial (except for the last number, which is the remainder).

Tips & Warnings

• You can compare this technique with long division of a polynomial by a binomial in the form of x-c.

Resources

• Practice using the synthetic division gizmo at Explore Learning.