How do you solve for x?
x^4 - 4x^3 + 3x^2 + 8x - 16 = 0

I tried using rational root theorem and synthetic division so many times but it won't work. I can't find any zeros. D: Am I simply doing it wrong or what? :(

Question

How do you solve for x? 
x^4 - 4x^3 + 3x^2 + 8x - 16 = 0

I tried using rational root theorem and synthetic division so many times but it won't work. I can't find any zeros. D: Am I simply doing it wrong or what? :(

anonymous · Answer

Try to take the first three terms $x^4-4x^3 +3x^2$ and factorize.

jhannybean · Answer

what if we .... $$\large x^4 - 4x^3 + 3x^2 + 8x - 16 = 0 $$$$\large x^4 - 4x^3 + 3x^2 + 8x =16$$$$\large x(x^3-4x^2+3x+8)=16$$

anonymous · Answer

I tried those as well but I always reach a dead end. :( Unless, I'm doing it wrong.

jhannybean · Answer

Somewhere we're going to have to use the Rational Zero Theorem perhaps?

campbell_st · Answer

well the quartic has 2 quadratic factors... thats why its hard to solve


you are looking at 

$$(x^2 -3x + 4)(x^2 -x - 4) = 0$$

now solve the quadratics for the solutions.

jhannybean · Answer

How did you do that @campbell_st O_o

anonymous · Answer

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

campbell_st · Answer

just practice... and a bit of luck

luigi0210 · Answer

aren't we gonna have some imaginary solutions?

jhannybean · Answer

Dx i meant what method of factorization did you use to.... solve this.

campbell_st · Answer

1 quadratic will the other has unequal irrational solutions.

anonymous · Answer

He used intuition. To factor in 2 terms, the largest orders had to be $x^2$

jhannybean · Answer

I see. I'm siding more towards @DanielM_113 's method. Interesting concepts you guys.

campbell_st · Answer

its just working backwards... and a little trial and error... given the rational root theorem only works for linear roots..

jhannybean · Answer

Ohh I see.

anonymous · Answer

I don't see. D: Can you explain a little more? Sorry.

campbell_st · Answer

I just see patterns... and work from there.... I saw a lot of 4s 

3 = 4 - 1   -4 = -2 x 2 or -1 x 4 -16 = -4 x 4

its just stuff like that...

anonymous · Answer

Oh :O Well, if I do more factoring practice will i be able to be like you... :'D

campbell_st · Answer

maths is very practical... to more you do the better you get

anonymous · Answer

okay, thanks :D hopefully, i will get this soon.

jhannybean · Answer

And helping others solve problems actually really helps in understanding these concepts,as well as practicing problems :D I've learned so many new techniques i never knew before.

anonymous · Answer

Do you know the Russian multiplication technique?

anonymous · Answer

Nope D:

anonymous · Answer

Russian multiplication @ineptAtMath http://openstudy.com/updates/51ad8bb2e4b06ee3ee25df07