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Synthetic Division

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

The result is

4*x*^{3} +6*x*^{2} + 6*x* + 8, and the remainder is 26.