Long division of a polynomial by a binomial is carried out in essentially the same manner as long division of two integers with no variables:

- Divide the highest degree term of the polynomial by the highest degree term of the binomial. Write the result above the division line.
- Multiply this result by the divisor, and subtract the resulting binomial from the polynomial.
- Divide the highest degree term of the remaining polynomial by the highest degree term of the binomial.
- Repeat this process until the remaining polynomial has lower degree than the binomial.

*Example*: Divide
2*x*
^{4} -9*x*
^{3} +21*x*
^{2} - 26*x* + 12
by
2*x* - 3
.

Figure %: Long Division

The following two theorems have applications to long division:

Remainder Theorem. When a polynomial
*P*(*x*)
is divided by
*x* - *a*
, the
remainder is equal to
*P*(*a*)
.

Factor Theorem. If
*P*(*x*)
is a polynomial and
*P*(*a*) = 0
, then
*x* - *a*
is a
factor of
*P*(*x*)
. In other words, if the remainder when
*P*(*x*)
is divided by
*x* - *a*
is 0, then
*x* - *a*
is a factor of
*P*(*x*)
.

*Example*: If
*P*(*x*) = 3*x*
^{3} -2*x*
^{2} + 4*x* - 1
, use the Remainder Theorem to
find the remainder when
*P*(*x*)
is divided by
*x* - 2
.

The remainder is 23.P(2) = 3(2)^{3}-2(2)^{2}+ 4(2) - 1 = 23 .

*Example*: Is
*x* + 3
a factor of
*P*(*x*) = *x*
^{4} +2*x*
^{3} -7*x*
^{2} + 2*x* - 8
?

Is
*x* - 2
a factor of
*P*(*x*) = *x*
^{4} +2*x*
^{3} -7*x*
^{2} + 2*x* - 8
?

ThusP(- 3) = (- 3)^{4}+2(- 3)^{3}-7(- 3)^{2}+2(- 3) - 8 = - 50≠ 0 .

P(2) = (2)^{4}+2(2)^{3}-7(2)^{2}+ 2(2) - 8 = 0 .

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