# Factor theorem

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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 $f\left(x\right)$ has a factor $x-k$ if and only if $f\left(k\right)=0$.

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 $\left(x - 1\right)$ a factor? To find out, substitute $x = 1$ into the polynomial above:: $x^3 + 7x^2 + 8x + 2 = \left(1\right)^3 + 7\left(1\right)^2 + 8\left(1\right) + 2$: $= 1 + 7 + 8 + 2$: $= 18$

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

This is equal to $0$. Therefore $x-\left(-1\right)$, 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 $\left(x+1\right)$ to get a quadratic, which can be solved directly, by the factor theorem or by the quadratic equation. $\left(x^3 + 7x^2 + 8x + 2\right) over \left(x + 1\right)$ = $x^2 + 6x + 2$ and therefore $\left(x+1\right)$ 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 $a in mathbb\left\{C\right\}$. Then $f\left(a\right) = 0$ iff $f\left(x\right)$ can be written in the form $f\left(x\right)=\left(x-a\right)g\left(x\right)$ where $g\left(x\right)$ is also a polynomial. $g$ is determined uniquely.

This indicates that those $a$ for which $f\left(a\right) = 0$ are precisely the roots of $f\left(x\right)$. 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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