Algebra II: Polynomials
Long Division of a Polynomial by a Binomial
Long Division of a Polynomial by a Binomial
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
2x
4 -9x
3 +21x
2 - 26x + 12
by
2x - 3
.
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) = 3x
3 -2x
2 + 4x - 1
, use the Remainder Theorem to
find the remainder when
P(x)
is divided by
x - 2
.
P(2) = 3(2)3 -2(2)2 + 4(2) - 1 = 23 .The remainder is 23.
Example: Is
x + 3
a factor of
P(x) = x
4 +2x
3 -7x
2 + 2x - 8
?
Is
x - 2
a factor of
P(x) = x
4 +2x
3 -7x
2 + 2x - 8
?
P(- 3) = (- 3)4 +2(- 3)3 -7(- 3)2 +2(- 3) - 8 = - 50≠ 0 .Thus x + 3 is not a factor of P(x) = x 4 +2x 3 -7x 2 + 2x - 8 , but x - 2 is a factor of P(x) .
P(2) = (2)4 +2(2)3 -7(2)2 + 2(2) - 8 = 0 .





