1

I got this polynomial equations of degree 4 $$x^{4}-6x^{3}-36x^2+216x-324=0$$ from Crossed Ladders Problem and i'm tired to solve it without using WF or any software calculator

I even read solution of Is there a general formula for solving 4th degree equations (quartic)? but looks like hard

Community
  • 1
Educ
  • 4,780
  • 1
    If you know that the roots are integers, you can make use of vietas formulas. E.g. For the case of degree 2 if the roots are $a$ and $b$, the polynomial is $(x-a)(x-b) = x^2-(a+b)x+ab$ so you know the constant term is the product of all four solutions, etc. In your case you have $-324 = -2^2\cdot 3^4$. There fore you know that one or three roots have negative coefficients, and the only ones possible, are divisors of $324$. – flawr Feb 01 '15 at 10:33
  • @flawr: But here the roots are not integers: there is one between $-7$ and $-6$ and another between $7$ and $8$. – Henry Feb 01 '15 at 10:41
  • Well then I do not think that there is an easier way of finding the roots, tan using the general formula from galois theory or any numerical methods (e.g. Newton's method ) – flawr Feb 01 '15 at 10:48
  • could you do it by using the Galois Theory – Educ Feb 01 '15 at 11:04
  • 2
    See quartic formula. – Lucian Feb 01 '15 at 11:15
  • @Educ - How sure are you that you derived the correct quartic equation above - it might be worthwhile stating the original problem in full. – Mufasa Feb 01 '15 at 12:41
  • @Mufasa see my update – Educ Feb 01 '15 at 16:26
  • @Educ This question now has an answer at https://math.stackexchange.com/questions/1124801/crossed-ladders-problem?noredirect=1&lq=1, just let $x=h_j$. – someone Apr 05 '19 at 14:11

0 Answers0