Factoring polynomials

[mathwannabe]

Hello everybody

I have identified my main weakness: factoring polynomials. I can factor, but it takes me more time and energy than I feel it should. I will give an example:

Factor: \(\displaystyle x^3-3x+2\)

\(\displaystyle x^3-3x+2=\)
\(\displaystyle =x^3-x-2x+2=\)
\(\displaystyle =x(x^2-1)-2(x-1)=\)
\(\displaystyle =x(x-1)(x+1)-2(x-1)=\)
\(\displaystyle =(x-1)(x(x+1)-2)=\)
\(\displaystyle =(x-1)(x^2+x-2)=\)
\(\displaystyle =(x-1)(x-1)(x+2)=\)
\(\displaystyle =(x-1)^2(x+2)\)

Now, I have a feeling that there must be a better way that is less relying on intuition. Please, I would be grateful for any hints on how to factor more easily.

EDIT: Some good online reference would be great

 

[lookagain]

Hello everybody

I have identified my main weakness: factoring polynomials. I can factor, but it takes me more time and energy than I feel it should. I will give an example:

Factor: \(\displaystyle x^3-3x+2\)

\(\displaystyle x^3-3x+2=\)
\(\displaystyle =x^3-x-2x+2=\)
\(\displaystyle =x(x^2-1)-2(x-1)=\)
\(\displaystyle =x(x-1)(x+1)-2(x-1)=\)
\(\displaystyle =(x-1)(x(x+1)-2)=\)
\(\displaystyle =(x-1)(x^2+x-2)=\)
\(\displaystyle =(x-1)(x-1)(x+2)=\)
\(\displaystyle =(x-1)^2(x+2)\)

Now, I have a feeling that there must be a better way that is less relying on intuition.
Please, I would be grateful for any hints on how to factor more easily.

EDIT: Some good online reference would be great

You could try and use the Rational Root Theorem:

http://en.wikipedia.org/wiki/Rational_root_theorem


And then you could use long or synthetic division to come up with a polynomial of
lesser degree to factor.

 

[mmm4444bot]

I have a feeling that there must be a better way that is less relying on intuition. Please, I would be grateful for any hints on how to factor more easily.

The factor-by-grouping method that you posted is a good way for such exercises.

Homework exercises are often contrived to be simple examples; in other words, the factors usually involve Integers that are near zero.

So, one thing to consider right away is whether any simple values of x (eg: 1, 2, -1, -2) cause the polynomial to evaluate to zero because such values of x lead to factors of the polynomial (google factor theorem for explanatory sites, like those at the link).

In your example exercise, you can tell by inspection that both x=1 and x=-2 lead to x^3-3x+2=0. Therefore, (x-1) and (x+2) must be factors. That Polynomial Division suggested by lookagain will give you remaining factors, one you deduce at least one factor.

Perhaps, this type of ansatz will save you a little time, when dealing with factorizations that involve simple examples.

Cheers ~ Mark :cool: