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
2 + 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
b(x - a)
and divide the polynomial by
. Otherwise, leave the
x - a
- Write the value of
, and write all the coefficients of the polynomial in
a horizontal line to the left of
- 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
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
terms, the first
n - 1
numbers below the line
are the coefficients of the resulting polynomial, and the last number is the
Example: What is the result when
2 - 10x + 2
x - 3
? What is the remainder?
Figure %: Synthetic Division
The result is
2 + 6x + 8
, and the remainder is 26.