CraigsMaths

There are several approaches to factoring polynomials.  Some are easier than others depending on the polynomial you are trying to factor.  Here’s a step by step method to help you on the way to selecting the “best” approach to use.  This article only considers real solutions and does not deal with complex solutions.

When to factor?

Factoring is used when:

1. Solving a polynomial equation
2. Finding the roots of a polynomial equation
3. Finding the zeros of a polynomial function
4. Factoring a polynomial function
5. Finding the x-intercepts of a polynomial function

In fact all five of these are the same thing and hence will use the following process.

How many factors?

An degree polynomial will have exactly factors and roots.

If the sign of the coefficients of each term changes times in then there will be or less an even number of positive roots.

If the sign of the coefficients of each term changes times in then there will be or less an even number of negative roots.

By knowing what to expect you will know where to start and what to look for when finding factors.

Step 1 - Arrange the function into the standard form

The standard form consists of the terms arranged in descending order of exponent with the coefficients expressed as integers.

Where is the degree of the polynomial.

Step 2 - Factorise on the common factors

Look at the polynomial you now have and find any common factors of all the terms.  This includes any common factors that have powers of in them.

For example the common factors of:

are and

So factorising this we get:

Step 3 - Look for standard forms and factorise them

Difference of squares

Where the expression can be expressed as a difference of two squares the factors are also determined directly:

Perfect squares

Sum of cubes

(Note the resulting quadratic will not factor.)

Difference of cubes

(Note the resulting quadratic will not factor.)

Grouping

For if then you can group terms and common factors will be apparent.

Step 3 - Test for rational factors

For any factors that have a degree greater than 1 you need to try to factorise further.  If these are quadratics then go to step 4.  Otherwise continue with this step.

First we need a short list.  We can get this from the rational roots test. First make sure that all your coefficeints are integers.  This should have happened at step 1 already. Make a list of possible rational roots by constructing fractions with the denominators formed from all the factors of the constant term and the numerator from all the factors of the leading coefficient.  List each +/- combination.

You can further eliminate numbers from this list by considering the sign of the roots that you expect.

It may turn out that none of these are roots - but they are worthy first candidates and no other rational numbers can possibly be roots.

If there are no rational roots you then need to look to the possibility of irrational and complex roots.

Check each of the possible rational roots by:

1. Long division
2. Synthetic division
3. OR Substitution

If the polynomial is large the computational overhead for 3 may be greater than 1 or 2 and not yeild as much useful results.  1 and 2 will give you the factors.

Step 4 - Factorise the quadratics

Factors of that add up to .  This will yield rational roots.

http://www.craigsmaths.com/quadratic-equations/factors-of-quadratics/

Quadratic formula.  This may yield non-rational, complex or rational roots.

Use the discriminant to determine the nature of factors.

Step 5 - Repeat

Repeat the above steps on each factor found so far until the equation can no longer be reduced.

Tags: factors, polynomials

Saturday, January 24th, 2009 at 1:32 am
Category: Engineering Maths B, Trinomial factors.
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