Long division is useful with the remainder and factor theorems, but long
division can be time consuming. To divide a polynomial by a binomial and
compute the remainder, we can also use synthetic division. We can only
divide by a binomial whose leading
coefficient is 1--thus, we must factor the
leading coefficient out of the binomial and divide by the leading coefficient
separately. Also, the binomial must have degree 1; we cannot use synthetic
division to divide by a binomial like x2 + 1. Here are the steps for
dividing a polynomial by a binomial using synthetic division:
- Write the polynomial in descending order, adding "zero terms" if an exponent term is skipped.
- If the polynomial does not have a leading coefficient of 1, write the binomial as b(x - a) and divide the polynomial by b. Otherwise, leave the binomial as x - a.
- Write the value of a, and write all the coefficients of the polynomial in a horizontal line to the left of a.
- Draw a line below the coefficients, leaving room above the line.
- Bring the first coefficient below the line.
- Multiply the number below the line by a and write the result above the line below the next coefficient.
- Subtract the result from the coefficient above it.
- Repeat steps 6 and 7 until all the coefficients have been used.
- If the polynomial has n terms, the first n - 1 numbers below the line are the coefficients of the resulting polynomial, and the last number is the remainder.
Example: What is the result when 4x4 -6x3 -12x2 - 10x + 2 is divided by x - 3? What is the remainder?
The result is 4x3 +6x2 + 6x + 8, and the remainder is 26.