If you write a polynomial as the product of two or more polynomials, you have **factored** the polynomial. Here is an example:

The polynomials *x*-3 and are called **factors** of the polynomial . Note that the degrees of the factors, 1 and 2, respectively, add up to the degree 3 of the polynomial we started with. Thus factoring breaks up a complicated polynomial into easier, lower degree pieces.

We are not completely done; we can do better: we can factor

We have now factored the polynomial into three **linear** (=degree 1) polynomials. Linear polynomials are the easiest polynomials. We can't do any better. Whenever we cannot factor any further, we say we have **factored** the polynomial **completely**.

Once again consider the polynomial

Let's plug in *x*=3 into the polynomial.

Consequently *x*=3 is a root of the polynomial .
Note that (*x*-3) is a factor of .

Let's plug in into the polynomial:

Thus, is a root of the polynomial . Note that is a factor of .

Since we have already factored

there is an easier way to check that *x*=3 and are roots of *f*(*x*), using the right-hand side:

Does this work the other way round?
Let's look at an example: consider the polynomial . Note that *x*=2 is a root of *f*(*x*), since

Is (*x*-2) a factor of ? You bet!
We can check this by using long polynomial division:

So we can factor

Let's sum up: Finding a root *x*=*a* of a polynomial *f*(*x*) is the same as having (*x*-*a*) as a linear factor of *f*(*x*). More precisely:

**Given a polynomial f(x) of degree n, and a
number a, then **

** **

** if and only if there is a
polynomial q(x) of degree n-1 so that **

** **

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Sun Jun 8 13:42:54 MDT 1997

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