problems of Roots of polynomials

**kin** · October 29th 2009, 05:42 AM

M^3 - M -1 =0

how to solve for M...?
i take approximation for M=1.33, am i right?

**proscientia** · October 29th 2009, 06:13 AM

Originally Posted by kin

M^3 - M -1 =0

how to solve for M...?
i take approximation for M=1.33, am i right?

There are many numerical methods you can use for finding approximations to the real root of the equation. Use the intermediate-value theorem to narrow it down first; in this case, the real root is between $1$ and $1.5.$ Then you can apply, for example, Newton’s method to the polynomial.

For an analytic solution, you can use Cardano’s method, which in this case yields an exact solution of $\sqrt[3]{\frac{1+\sqrt{\frac{23}{27}}}2}+\sqrt[3]{\frac{1-\sqrt{\frac{23}{27}}}2}$ for the real root (which is approximately $1.3247).$

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