Question

Use the rational zero theorem, Descartes rule of signs, and the theorem on bounds as aids in finding all real and imaginary roots to the following equation:
4x^(3)-17x^(2)+16=0

Answer 1

By the Descartes rule of signs, we know that there are two positive roots out of three, which also tells us that all the roots are real.

Using the rational zero theorem, we know that rational roots, if any, have to be of the form ±p/q, where p is a factor of 16, and q is a factor of 4.

So the possible rational zeroes are:
±(1/4,1/2,1,2,4,8 and 16), from which we can find x=4 is a root.

Use polynomial division to reduce the polynomial to a quadratic from which the two remaining (irrational) roots can be found.

Note: If we differentiate the polynomial and equate to zero, we find that the local maximum/minimum are at 0 and 2.8. The theorem on bounds tells us that there is a zero between 0 and 2.8, and the other two are at <0 and >2.8 respectively.