B What are the different methods of factoring polynomials?

1. Jul 1, 2017

awholenumber

Methods of factoring .

Method of common factors
Factorization by regrouping terms
Factorization using identities
Factors of the form ( x + a) ( x + b)
Factor by Splitting

Is this all the factoring methods out there ? Or are there more ?

I am also looking for a book with lots of practice problems of factoring polynomials .

2. Jul 1, 2017

SlowThinker

If you want to factorize school examples that are designed to have nice results, you can use whatever tricks come to your mind.
If you want to do it in practice, you find the roots first, either with a formula (probably only with quadratic polynomials) or using one of several iterative methods (I like Laguerre's method but there are dozens).
Once you have the roots, say a, b, c, d then the factorization is $k(x-a)(x-b)(x-c)(x-d)$. Obviously the number of factors depends on the degree of the polynomial.

Edit: forgot to put in the constant multiplier $k$ that is the same as the coefficient of the highest degree, e.g. 5 in $5x^3-2x+1$.

3. Jul 1, 2017

awholenumber

One reason why i like this forum is that you always get a fast reply no matter how stupid your questions are .

:-)

4. Jul 1, 2017

I like Serena

You may like the Rational root theorem.

If we can split a polynomial with integer coefficients, like $3x^3 - 5x^2 + 5x - 2$, in the form $a(x-b)(x^2+...)$ where $b$ is an integer or a rational number, then $b$ has to be plus or minus a divider of $2$ divided by a divider of $3$.
That is, b is one of $\pm \frac 11, \pm\frac 21, \pm\frac 13, \pm \frac 23$.
And indeed, if we substitute $x=\frac 23$, we find that it's a root.
So we can split it as $3(x- \frac 23)(x^2 + ..)$ or more cleanly as $(3x- 2)(x^2 + ..)$.

5. Jul 1, 2017

awholenumber

Currently i am only following the materials available from these resources .

http://ncert.nic.in/textbook/textbook.htm
https://2012books.lardbucket.org/books/beginning-algebra/
http://ins.sjtu.edu.cn/people/mtang/textbook.pdf

Most of those resources don't have factorization of Cubic polynomials in it , Don't know why though .
I guess those methods are more advanced than the usual methods .

I am only getting comfortable with explanations like these ,

A whole number has "pairs of factors" in it which when multiplied together gives you the whole number
I like that explanation because it says "pairs of factors" , you can then chose the "pairs of factors" that goes to the middle part of the polynomial when factoring it .

I am only trying to find more examples of "Factor by Splitting"

:-)

6. Jul 1, 2017

Staff: Mentor

Because factoring a cubic polynomial is generally very complicated. There are a few cubics that have nice factorizations, such as $(x^3 + a^3) = (x + a)(x^2 - ax + a^2)$ and $(x^3 - a^3) = (x - a)(x^2 + ax + a^2)$.

7. Jul 2, 2017

awholenumber

Thanks for the explanations :-)