Factor theorem

Short description: A polynomial has a factor (x-k) if and only if k is a root

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. 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 $f(k)=0$ (i.e. $k$ is a root).

Factorization of polynomials

Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.

The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:

1. "Guess" a zero $a$ of the polynomial $f$. (In general, this can be very hard, but maths textbook problems that involve solving a polynomial equation are often designed so that some roots are easy to discover.)
2. Use the factor theorem to conclude that $(x-a)$ is a factor of $f(x)$.
3. Compute the polynomial $g(x)=f(x)/(x-a)$, for example using polynomial long division or synthetic division.
4. Conclude that any root $x\ne a$ of $f(x)=0$ is a root of $g(x)=0$. Since the polynomial degree of $g$ is one less than that of $f$, it is "simpler" to find the remaining zeros by studying $g$.

Example

Find the factors of

$x^3+7x^2+8x+2.$

To do this one would use trial and error (or the rational root theorem) to find the first x value that causes the expression to equal zero. To find out if $(x-1)$ is a factor, 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 and not 0. This means $(x-1)$ 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:

$\frac{x^3+7x^2+8x+2}{x+1}=x^2+6x+2,$

and therefore $(x+1)$ and $x^2+6x+2$ are factors of $x^3+7x^2+8x+2.$ Of these the quadratic factor can be further factored using the quadratic formula, which gives as roots of the quadratic $-3\pm \sqrt{7}.$ Thus the three irreducible factors of the original polynomial are $x+1,$ $x-(-3+\sqrt{7}),$ and $x-(-3-\sqrt{7}).$