1. ## solving polynomials

How would I solve and equations like it: $\displaystyle d^3 - \frac{1}{d} = 5.6023$ without graphing calcs.

I am working on finding the diameters of hollow shafts when given allowable stresses and I wind up with these eqautions.

Thanks for any help.

2. That's not a polynomial, it doesn't have positive integer exponents (because $\displaystyle \frac{1}{d} = d^{-1}$)

Otherwise I have no idea how to solve this, I just wanted to point that out.

3. ## unsimplified

$\displaystyle d^4 - 5.6023d - 1 = 0$

4. I dunno if it is actually possible in an easy way ... you can't reduce the polynomial and it's a 4th degree one ... hard ... you may need to apply Ferrari's or Euler's method to do that !

5. In general, there is no easy way to solve polynomials. The polynomial here is a quartic, which can be solved (although not easily) following the approach found here:
Quartic function - Wikipedia, the free encyclopedia

A similar approach can solve quintics (polynomials of degree 5) but there is no general method for solving higher order polynomials

6. Linear and quadratics are easy because mathematicians have found a highly efficient and easy way to solve them. Cubic polynomials are quite harder (but still solvable in most situations), quartics get hard and quintics even harder (wanna try elliptic curves ?). And for higher degree ... well badgerigar sait it all