## zerlinav 3 years ago What are the possible number of positive, negative, and complex zeros of f(x) = 3x4 - 5x3 - x2 - 8x + 4

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1. richyw

Ok well first of all it's a quadratic function, so the total will be 4. Use Descartes’ Rule of Signs. Your function is $p(x)=3x^4-5x^3-x^2-8x+4$ So count the number of sign changes. A "sign change" is when the sign the coefficient of the $$a^n$$ is different from the sign of the coefficient of $$a^{n-1}$$. So here you have one sign changes. You do the same thing for $$p(-x)$$ and also find one sign change (don't include the sign change for the 4! only the ones that multiply by x matter. Since there must be 4 total you know that it is one positive, one negative and two complex (the complex ones have to be an even number!)

2. richyw

sorry not quadratic. haha. 4th degree polynomial

3. richyw

4. cruffo

I thought there were two sign changes. $p(x)=3x^4-5x^3-x^2-8x+4 \rightarrow \color{red}{+}\color{red}{-}- \color{red}- \color{red}+$

5. cruffo

# of possible positive zeros is the number of sigh changes of the polynomial, or less than that number by an even integer. So the number of possible positive zeros is 2 or 0

6. richyw

there is only one sign change though. That last one doesn't matter.

7. cruffo
8. richyw

weird, I might be wrong. But I don't think so.

9. cruffo

Take a look at the graph: http://www.wolframalpha.com/input/?i=p%28x%29%3D3x^4-5x^3-x^2-8x%2B4+from+-3+to+3

10. richyw

oh yeah you are totally correct!

11. richyw

so in this case using the rule of signs all that you can know is one of four possible combinations then right? Also I have been doing this wrong for like 5 years haha. good thing we have calculators...

12. cruffo

: ) Sounds right: degree 4 means you will find 4 solutions to p(x) = 0 For this polynomial, just looking at the signs, we can tell that there will be 2 or 0 positive roots, 2 or 0 negative roots. For the complex roots - they always come in pairs. so... case 1: 2 + real zeros, 0 - real zeros, and 2 complex zeros case 2: 0 + real zeros, 2 - real zeros, and 2 complex zeros case 3: 2 + real zeros, 2 - real zeros, and 0 complex zeros case 4: 0 + real zeros, 0 - real zeros, and 4 complex zeros