Factoring polynomials is an action in which a polynomial or polynomials are represented as a product of simpler polynomials that no longer can be factored.
Some factored polynomials you know from before:
and so on.
How do you factorize? This is very simple, you just take the common multiple of all your terms and take them out in front of braces.
Example 1: Factorize: x^2 + 4x. Their common multiple is obviously x so we’ll extract them in front of braces, and multiply by a polynomial divided by x.
How do you check it? Simply multiply those two factors you got:
Example 2: Factor the following polynomial
Here we can factor from every term. . Here we got a quadratic polynomial which we can factorize using its zeros. We got that zeros of this quadratic polynomial are -3 and 2. Now we can write the final factorized polynomial.
What are the zeros and their multiplicity? We have first factor x^2, its zero is 0 with multiplicity 2, zero of (x + 3) is -3 , and zero of (x – 2) is 2 both with multiplicity 1. Remember that the number of all these zeros must add up to the leading exponent of given polynomial.
Factorize . The common multiple of these terms is xy. . If you ever get unsure about what is your full common multiple you can always solve it in few steps, first extract x to simplify, and then you’ll easily notice more common multiples.
Example 5: Factorize . Here the factors aren’t seeable at first, but you can use all your knowledge you have to solve this.
You already know that . And our polynomial can be written like this: . In this case , and . So it is easy to conclude that
But what with cases you can’t conclude the solution so easily?
Example 6: Factorize .
You do this by grouping certain terms, and factorize them individually and then trying to factorize whole polynomial:
And now you should notice that your polynomial has two terms that have one factor the same, so you can extract him, and get your factor:
You could also group differently:
There is another way in factorizing. We’ll show it on quadratic polynomial (whose largest exponent equals 2 ). That is finding zeros.