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
*x*
^{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
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
4*x*
^{4} -6*x*
^{3} -12*x*
^{2} - 10*x* + 2
is
divided by
*x* - 3
? What is the remainder?

Figure %: Synthetic Division

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