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I have tried to solve this 8th degree equation & calculated one approximate real root using Newton-Raphson method it is $x=1.340775827$. How to find other real roots of this equation $$60x^8+252x^7-1061x^6-4x^5+647x^4+500x^3+149x^2+20x+1=0$$

Any help is greatly appreciated. Thanks!

AlexR
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Harish Chandra Rajpoot
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    If you had found an exact root, polynomial long division by $(x-x_0)$ would give a polynomial with the found root removed. – AlexR Mar 11 '15 at 10:06
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    This factors as a product of two irreducible quartics, so all roots can be extracted using roots. See, e.g., http://math.stackexchange.com/questions/785/is-there-a-general-formula-for-solving-4th-degree-equations – Travis Willse Mar 11 '15 at 10:11
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    http://www.wolframalpha.com/input/?i=60x%5E8%2B252x%5E7-1061x%5E6-4x%5E5%2B647x%5E4%2B500x%5E3%2B149x%5E2%2B20x%2B1%3D0 – Emilio Novati Mar 11 '15 at 10:12
  • AlexR's idea of doing long division is great. Of course, this works best when you have an exact root. A numerical result may not be factor quite evenly if its not exactly right. Although this might not be a problem for you. – sav Mar 11 '15 at 10:13
  • But, I could calculate only one approximate real root. How to find the exact real root which is approximate 1.340775827... – Harish Chandra Rajpoot Mar 11 '15 at 10:13
  • Also check the Able impossibility theorem, http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem Its not always possible to get an exact algebraic solution. – sav Mar 11 '15 at 10:36
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    To get the exact value enter 10*x^4-33*x^3+19*x^2+9*x+1)=0 into Wolfram Alpha and press exact forms. This gives you a very complicated expression with the approximate value of $\approx 1.3408.$ The given quartic is one factor of your original polynomial. – gammatester Mar 11 '15 at 10:37
  • Did you also want the imaginary roots? – sav Mar 11 '15 at 10:55
  • @gammatester how did you find this factor? – Oscar Lanzi Jan 28 '22 at 21:26

2 Answers2

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How to find roots of 8th degree equation

The best way to find something is by not looking for it in the first place. Instead, just try and factor the polynomial, to see if by any chance it cannot be written as a product of two smaller ones, of lesser degree, whose roots might therefore be easier to find. Indeed, it can be written as the product of two quartics, meaning that its roots possess an exact closed form expression:

$$P_8(x)=\Big(1+11x+31x^2+45x^3+6x^4\Big)~\Big(1+9x+19x^2-33x^3+10x^4\Big).$$

Lucian
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The link posted by Emilio shows the shape of the curve. This show you where to choose your starting points for the newton-raphson method.

Different starting points for newton raphson will lead to different solutions. So pick starting points near the solutions shown on the graph.

sav
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