Factoring a very long expression

[rygza]

Factoring a very long expression! please help

(x^6) - (4x^5) + (x^4) + (10x^3) - (4x^2) - (8x)

then:

x((x^5) - (4x^4) + (x^3) + (10x^2) - (4x) - 8)

my teacher skipped many steps and ultimately found:
x * [(x+1)^2] * [(x-2)^3]

I guess he's assuming we know how to get through it. Can you guys please guide me through how to find the factored form? I've been looking through my old pre-calc book but i don't think we ever covered this.

(BTW, this isn't the homework problem, i just need to find how to factor it because i then use the factored form for solving homogeneous linear ODEs)

 

[issacnewton]

use the rational roots test to guess at the roots of the polynomial with degree 5. there the constant term is -8 and the leading coefficient is 1. so according to the rational roots test

http://www.purplemath.com/modules/solvpoly2.htm
http://www.purplemath.com/modules/rtnlroot.htm

the possible roots are \pm 1,\pm 4,\pm 2,\pm 8

now once you have guessed the roots, try synthetic division to check them

 

[rygza]

use the rational roots test to guess at the roots of the polynomial with degree 5. there the constant term is -8 and the leading coefficient is 1. so according to the rational roots test



the possible roots are \pm 1,\pm 4,\pm 2,\pm 8

now once you have guessed the roots, try synthetic division to check them

ok, got how to find the roots. factors of const. coefficient over factors of leading coefficient.
how do i check them with synthetic division?

 

[rygza]

use the rational roots test to guess at the roots of the polynomial with degree 5. there the constant term is -8 and the leading coefficient is 1. so according to the rational roots test



the possible roots are \pm 1,\pm 4,\pm 2,\pm 8

now once you have guessed the roots, try synthetic division to check them

i tried it out and 0,-1, and 2 are indeed roots. but how do i know that -1 is a root twice, and 2 is a root twice? from the most factored form, how does one know that it is (x+1) squared and (x+2) cubed?

 

[HallsofIvy]

Since 0, -1, and -2 are roots, you know that there are factors of x, x+1, and x+ 2. Divide the original polynomial by those and see what is left. What are the roots of that?

 

[issacnewton]

rygza, follow what HallsofIvy said.