How to solve an equation where the x value is bigger than x^2?

Question

ahsome · Answer

Example being:$$2x^4+3x^3-x^2+9=0$$

ahsome · Answer

You cannot use quadratic equation. One of the roots will be given

dumbcow · Answer

this occurs when 0<x<1

ahsome · Answer

Ok then. How would you factorise this equation?

dumbcow · Answer

hmm im not sure that it does factor

usually these may be solved by graphing and/or synthetic division to factor out roots

ahsome · Answer

I have learnt that if I am given one of the roots, I would have to divide it by that root using polynomial division. Then I would have to use T&E to find another root.

BTW, The equation was an example I just made up, not sure if it even works ;)

jtvatsim · Answer

Great question. In the history of mathematics, finding a general solution process for x^3 and x^4 polynomials was very difficult. For example, there is a "trinomial equation" and a "quatronomial equation" (I'm making the exact names up), but they are extremely messy and complicated. So, for most of us, using roots and T&E are the only reasonable methods.

jtvatsim · Answer

As an interesting fact, it has been mathematically proven that it given a general polynomial with x^5 and higher powers, it is impossible to solve with algebra techniques. An approximate solution is the best we can do.

ahsome · Answer

Ok, thanks. Do you know a good way to do T&E? I sadly don't have time in tests to do all possible combinations

anonymous · Answer

If I remember correctly, I had the same question and I found that in one YouTube video someone mentioned that, if there are integer roots, then they will occur at (plus, minus) the factors of the last coefficient.

Simple example:
2x^2 + 8x - 10

Factors of last are: +-(10, 1, 5, 2)
->One happens to be at -5, so you know that one factor is: (x+5).

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Slightly more complex example:
x^3+4x^2-11x-30

Factors of last are: +-(30, 1, 10, 3, 5, 6)
->One happens to be at +3, so you know a factor is: (x-3).
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It is again worth mentioning that this will only find the roots if at least one factor is in the form:
(x +- n)
or
(ax +- b)   --- Where b/a is an integer.

Hope this helps a bit. :)

ahsome · Answer

Oh, that's cool. Thanks :D