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**.

An intimately related concept is that of a **root**. also called a **zero**. of a polynomial. A number *x* =*a* is called a **root** of the polynomial *f* (*x* ), if

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 .

This is no coincidence! ** When an expression (x –a ) is a factor of a polynomial f (x ), then f (a )=0** .

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

Write down a polynomial with roots *x* =1, *x* =2, and *x* =3/4.