1. Jul 15, 2011

mathfriends

[URL]http://www13.0zz0.com/2011/07/15/22/754178270.jpg[/URL]

I Need help solving this fourth degree equation

Last edited by a moderator: Apr 26, 2017
2. Jul 15, 2011

Pengwuino

"I want answers" is not how you're going to get results. Read the rules of the forums.

3. Jul 15, 2011

micromass

Staff Emeritus
4. Jul 15, 2011

pmsrw3

Me, too.

5. Jul 15, 2011

Dr. Seafood

lol comic sans

6. Jul 16, 2011

mathfriends

thank you

Finally I found a solution

7. Jul 16, 2011

I like Serena

Hmm.
I understand the solution x=-1.
But how did you get the other solutions from $9x^3-4x+1=0$?

8. Jul 16, 2011

HallsofIvy

Staff Emeritus
The simplest way to approach this problem is to use the "rational root theorem".

If r is a rational number satisfying $a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0= 0$ where the coefficients are all integers, then r= p/q where p is an integer evenly dividing $a_0$ and q is an integer evenly dividing $a_n$.

Of course, it is not necessary than such an equation have any rational roots but it is worth trying. Here, the leading coefficient, $a_n$, is 9, which has factors $\pm 1, \pm 3, \pm 9$ and the constant term, $a_0$ is 1, which has factors $\pm 1$ so the only possible rational roots are $\pm 1, \pm 1/3, \pm 1/9$.

Putting those into the equation, we see that if
$$9(1)^4+ 9(1)^3- 4(1)^2- 3(1)+ 1= 19- 7= 12\ne 0$$
$$9(-1)^4+ 9(-1)^3- 4(-1)^2- 3(-1)+ 1= -9+ 9- 4+ 3+ 1= -1+1= 0$$
$$9(1/3)^4+ 9(1/3)^3- 4(1/3)^2- 3(1/3)+ 1= 1/9+ 1/3- 4/9- 1+ 1= 1/3- 1/3- 1+ 1= 0$$
We can stop here. Seeing that x= -1 and x= 1/3 are roots, we can divide by x+ 1 and x- 1/3 (not "0.3333") to get a quadratic equation that we can solve using the quadratic formula.

(There is a "quartic formula", http://www.sosmath.com/algebra/factor/fac12/fac12.html, but it is extermely complicated.)

9. Jul 16, 2011

I like Serena

Nice! I didn't know this one yet!

10. Jul 16, 2011

mathfriends

this is another solution

Is this true ?

11. Jul 16, 2011

Redbelly98

Staff Emeritus
Moderator's note: thread moved from "General Math".

Thank you.

12. Jul 16, 2011

Redbelly98

Staff Emeritus
Why don't you tell us? Are those answers the same as the ones you already know:

13. Jul 16, 2011

mathfriends

yeah,I just want clarification

14. Jul 16, 2011

I like Serena

What do you think?
Can you think of a way to derive this solution?

Btw, are you familiar with solving a quadratic equation?
That is, an equation of the form ax2 + bx + c = 0.

15. Jul 16, 2011

mathfriends

Yes, do you mean you want the law of quadratic equation

16. Jul 16, 2011

I like Serena

What I would like is for you to give us some insight into what you are thinking.

As it is we have no clue how you got this problem or why you want to solve it.
I'm assuming you're supposed to learn how to solve such equations.

How did you get this problem?
And what is your purpose with it?

17. Jul 16, 2011

mathfriends

thank you ( i like serena ) for help me .

18. Jul 17, 2011

keyfob

Dude! This is an awesome theorem, thanks for that.