Factor theorem

Factor theorem

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. x^2 + 6x + 6). It is a special case of the polynomial remainder theorem.

The factor theorem states that a polynomial f(x) has a factor x-k if and only if f(k)=0.

An example

You wish to find the factors of: x^3 + 7x^2 + 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:: x^3 + 7x^2 + 8x + 2 = (1)^3 + 7(1)^2 + 8(1) + 2: = 1 + 7 + 8 + 2: = 18

As this is equal to 18—not 0—(x - 1) is not a factor of x^3 + 7x^2 + 8x + 2. So, we next try (x + 1) (substituting x = -1 into the polynomial):: (-1)^3 + 7(-1)^2 + 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 x^3 + 7x^2 + 8x + 2.

The next two roots can be found by algebraically dividing x^3 + 7x^2 + 8x + 2 by (x+1) to get a quadratic, which can be solved directly, by the factor theorem or by the quadratic equation. (x^3 + 7x^2 + 8x + 2) over (x + 1) = x^2 + 6x + 2 and therefore (x+1) and x^2 + 6x + 2 are the factors of x^3 + 7x^2 + 8x + 2.

Formal version

Let f be a polynomial with complex coefficients, and a in mathbb{C}. Then f(a) = 0 iff f(x) can be written in the form f(x)=(x-a)g(x) where g(x) is also a polynomial. g is determined uniquely.

This indicates that those a for which f(a) = 0 are precisely the roots of f(x). Repeated roots can be found by application of the theorem to the quotient g, which may be found by polynomial long division.

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