Factor Theorem
Lesson · Beginner
Algebra: Polynomials
If P(x) is a polynomial and P(r)=0, then (x-r) is a factor of P(x).
To factor a polynomial of degree at least 3, one useful approach is to find a root r first. Then (x-r) is a factor, allowing further factorization.
By the Polynomial Division Algorithm, when P(x) is divided by x-r,
where R is a constant.
Substitute x=r
Then
so
Therefore,
if and only if
which happens if and only if x-r is a factor of P(x).
Hence,
The remainder when P(x) is divided by x-r is P(r).
If
for distinct r_i, then
r is a root of multiplicity at least m,
if and only if
If r is a root of exactly multiplicity m, then
but
Factor completely:
Try small integer values.
Therefore, x-1 is a factor.
Dividing gives
Then
Hence,
A cubic polynomial P(x) has roots 1, 2, 4 and P(0)=16.
Find P(x).
By the Factor Theorem,
Using P(0)=16 gives
so k=-2
Therefore,
Suppose
is divisible by both x-1 and x+2. Find a and b.
Since x-1 is a factor, P(1)=0
so
Since x+2=x-(-2) is a factor, P(-2)=0
so
Using a+b=-7 and 2a-b=1 gives
so a=-2 and b=-5
From P(-3)=0, the factor is x+3, not x-3.