Factoring and Roots of Polynomials

What is factoring?

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 tex2html_wrap_inline102 are called factors of the polynomial tex2html_wrap_inline104 . 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.

Roots of polynomials.

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 tex2html_wrap_inline104 . Note that (x-3) is a factor of tex2html_wrap_inline104 .

Let's plug in tex2html_wrap_inline126 into the polynomial:


Thus, tex2html_wrap_inline126 is a root of the polynomial tex2html_wrap_inline104 . Note that tex2html_wrap_inline132 is a factor of tex2html_wrap_inline104 .

Roots and factoring.

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 tex2html_wrap_inline126 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 tex2html_wrap_inline160 . Note that x=2 is a root of f(x), since


Is (x-2) a factor of tex2html_wrap_inline160 ? 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


Exercise 1.

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


Exercise 2.

Write down a polynomial with roots x=-1, x=3, and x=0.


Exercise 3.

Factor the polynomial tex2html_wrap_inline200 . Note that f(-2)=0.


Exercise 4.

Factor the polynomial tex2html_wrap_inline204 completely. Note that tex2html_wrap_inline206 and tex2html_wrap_inline208 .


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Helmut Knaust
Sun Jun 8 13:42:54 MDT 1997

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