Synthetic Division

Shortcut way to divide polynomials

Uses the roots (not factor form) as the divisor

Uses the coefficients of the terms, and not the variables, as the dividend

1. Change the factor form of the divisor to the root. (factor form is (x - 3), root is 3)

2. Write out the polynomial to be divided in descending form, writing in zeros for the terms that are missing. For example: x⁴ + 2x² + 4 would be written as: x⁴ + 0x³ + 2x² + 0x + 4

3. Write out the coefficients only of the polynomial, including signs and including the 0's

4. Write the root to the left of the coefficients with a vertical bar between it and the coefficients and a horizontal line underneath the coefficients (like an upside-down division symbol)

5. Bring the first coefficient down below the line, without doing anything to it

6. Multiply the coefficient by the root, and add the number to the next term.

7. Multiply the sum by the coefficient and add to the next term.

Repeat this process until there are no more terms.

If the remainder is 0, then the factor is a true root

If the remainder is not 0, then the value of the remainder is the value of the polynomial evaluated at the x-value of the divisor (what you would get if you substituted the divisor into the polynomial)

Note: You can only use this form if your divisor is a binomial

Always remember to put in the missing terms with a zero coefficient

The remaining number will start with a variable exponent of one less than the original. In the example to the right, the polynomial was 3rd degree, the quotient is 2nd degree.