- Factor theorem
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. ). It is a special caseof the polynomial remainder theorem.
The factor theorem states that a polynomial has a factor
if and only if.
You wish to find the factors of:
To do this you would use trial and error finding the first factor. When the result is equal to , we know that we have a factor. Is a factor? To find out, substitute into the polynomial above:: : :
As this is equal to 18—not 0— is not a factor of . So, we next try (substituting into the polynomial)::
This is equal to . Therefore , which is to say , is a factor, and -1 is a root of
The next two roots can be found by algebraically dividing by to get a quadratic, which can be solved directly, by the factor theorem or by the
quadratic equation. = and therefore and are the factors of
Let be a polynomial with complex coefficients, and . Then
iffcan be written in the form where is also a polynomial. is determined uniquely.
This indicates that those for which are precisely the roots of . Repeated roots can be found by application of the theorem to the quotient , which may be found by
polynomial long division.
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