I need help on how to factorize the following w/o calculator if possible.

$\displaystyle x^4+5x^3+2x^2+x-3=0$

I've attempted to factor with whole numbers with no avail.

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- Mar 30th 2010, 12:24 AMCthul[Factorization] Factorization of polynomials
I need help on how to factorize the following w/o calculator if possible.

$\displaystyle x^4+5x^3+2x^2+x-3=0$

I've attempted to factor with whole numbers with no avail. - Mar 30th 2010, 01:56 AMHallsofIvy
- Mar 30th 2010, 02:11 AMCthul
Okay.

I've checked the answers and the factor of it is.

$\displaystyle (x^2+x+1)(x^2+4x-3)=0$

For the solution to be true

$\displaystyle x=-2 \pm \sqrt {7}$

From what I know, via the answers.

$\displaystyle (x^2+x+1)=0$

Has no solutions because

$\displaystyle \Delta=-3$

$\displaystyle \Delta<0$

Then there are no solutions.

So, is there no way to do this mentally or by hand? Assuming I am factorizing over irrational numbers. - Mar 30th 2010, 02:58 AMmr fantastic
HOI reasonably assumed that you wanted linear factors.

Since linear factors involving integers are not possible, it's natural to then try to factor it as a product of two quadratics:

(x^2 + ax + b)(x^2 + cx + d)

and then see if you can find integer values of a, b, c and d. - Mar 30th 2010, 03:16 AMCthul
Oh, I see. Thanks.

- Apr 1st 2010, 06:28 PMCthul
I'm still stuck on this equation. I tried factorizing by identical equations but I end up with too many variables, is there really no way to factor this? I want to know how to factorize this.

(My attempt to factorize)

The identity?

$\displaystyle (x^2+ax+b)(x^2+cx+d)=0$

Expanded.

$\displaystyle x^4+ax^3+cx^3+acx^2+bx^2+dx^2+adx+bcx+bd=0$

Common Factor.

$\displaystyle x^4+x^3(a+c)+x^2(ac+b+d)+x(ad+bc)+bd=0$

So

$\displaystyle 5=a+c$

$\displaystyle 2=ac+b+d$

$\displaystyle 1=ad+bc$

$\displaystyle -3=bd$

Original Equation:

$\displaystyle x^4+5x^3+2x^2+x-3=0$ - Apr 8th 2010, 11:18 PMCthul
So does anyone have a method to solve this?

- Apr 8th 2010, 11:57 PMmr fantastic
- Apr 8th 2010, 11:58 PMCthul
It's an exercise from the book, and the section specified that calculators are not to be used.

- Apr 9th 2010, 12:15 AMBacterius
Hello,

you might want to solve the system for $\displaystyle a, b, c$ and $\displaystyle d$, thus recovering the factorized expression of the polynomial and solving it in the standard way.

However I don't know if it is possible to solve this system. You have four unknowns and four equations, but how to solve it I have no idea yet (haven't really looked). Try to work out some algebra around the system and see what you can do ?