Australia Finance

Oct 17 2017

If you write a polynomial as the product of two or more polynomials, you have

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 (xa ) 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 (xa ) 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.

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