Factor theorem
From Wikipedia, the free encyclopedia
In algebra, the factor theorem is a theorem for finding out the factors of a polynomial (an expression in which the terms are only added, subtracted or multiplied, e.g. x2 + 6x + 6). It is a special case of the polynomial remainder theorem.
[edit] An example
You wish to find the factors of
- x3 + 7x2 + 8x + 2
To do this you would use trial and error finding the first factor. When the result is equal to 0, we know that we have a factor. Is (x − 1) a factor? To find out, substitute x = 1 into the polynomial above:
- (13) + 7(12) + 8(1) + 2
This is equal to 18 not 0 so (x − 1) is not a factor of x3 + 7x2 + 8x + 2. So, we next try (x + 1) (substituting x = − 1 into the polynomial):
- ( − 13) + 7( − 12) + 8( − 1) + 2
This is equal to 0. Therefore x − ( − 1), which is to say x + 1, is a factor, and -1 is a root of x3 + 7x2 + 8x + 2
The next two roots can be found by algebraically dividing x3 + 7x2 + 8x + 2 by (x + 1) to get a quadratic, which can be solved directly, by the factor theorem or by the quadratic equation. 
= x2 + 6x + 2 and therefore (x + 1) and x2 + 6x + 2 are the factors of x3 + 7x2 + 8x + 2
[edit] Formal version
More formally, it states that for any polynomial
- f(x),
for all values of a which satisfy
- f(a) = 0,
(in which the value of a is substituted for x into the "y=" equation)
(x − a) is a factor of f(x). Or, more concisely:
is a polynomial.
This indicates that any a for which f(-a) = 0, is a root of f(x). Double roots can be found by performing polynomial long division.
