SOLUTION: This problem is not from my textbook it is a worksheet problem. Use Descartes’s rule of signs to discuss the possibilities for the roots of the equation. 4x3-9x2+2x+3=0

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: This problem is not from my textbook it is a worksheet problem. Use Descartes’s rule of signs to discuss the possibilities for the roots of the equation. 4x3-9x2+2x+3=0       Log On


   

Question 127623: This problem is not from my textbook it is a worksheet problem.
Use Descartes’s rule of signs to discuss the possibilities for the roots of the equation.
4x3-9x2+2x+3=0
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
4x%5E3-9x%5E2%2B2x%2B3=0
The signs are + - + +. So there are 2 sign changes, 1st term to 2nd term, and 2nd term to 3rd term.

Two sign changes means that there are at most 2 positive roots. There could also be 0 positive roots.

Evaluate f%28-x%29
4%28-x%29%5E3-9%28-x%29%5E2%2B2%28-x%29%2B3=0
-4x%5E3-9x%5E2-2x%2B3=0

The signs are - - - +. So there is 1 sign change from the 3rd to the 4th term.

One sign change on f%28-x%29 means that there is exactly 1 negative real root.

So overall, this equation has either 2 positive real roots and 1 negative real root, or a conjugate pair of complex roots and 1 negative real root.

A graph of the function demonstrates that the first possibility is the actual case.

graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C4x%5E3-9x%5E2%2B2x%2B3%29