## anonymous one year ago Please Help How could you use Descartes' rule and the Fundamental Theorem of Algebra to predict the number of complex roots to a polynomial as well as find the number of possible positive and negative real roots to a polynomial? Your response must include: A summary of Descartes' rule and the Fundamental Theorem of Algebra. This must be in your own words. Two examples of the process Provide two polynomials and predict the number of complex roots for each. You must explain how you found the number of complex roots for each.

1. anonymous

You can use Descarte's Rule of Signs to find the maximum number of positive and negative real zeros to a polynomial. The Fundamental Theorem of Algebra can be use to find the maximum number of total zeros. Then subtract the number of real zeros from the total zeros to obtain the complex zeros. I dont know what else to do.

2. jdoe0001

you could start by checking your books for both https://www.youtube.com/watch?v=5YAmwfT3Esc <-- descartes rule of signs http://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html <-- fundamental theorem of algebra

3. anonymous

You can use Descarte's Rule of Signs to find the maximum number of positive and negative real zeros to a polynomial. The Fundamental Theorem of Algebra can be use to find the maximum number of total zeros. Then subtract the number of real zeros from the total zeros to obtain the complex zeros. For example, taking the equation 7(x^5)-3(x^2)-9x+2. This equation has a total of two sign changes. This means the real postive roots are 2,0. Another example is -4(x^5)-5(x^2)+8x+2, There are 1 sign change in this getting a negative real root, and the only negative real root is 1.

4. anonymous

Is this good?

5. anonymous

@jdoe0001

6. anonymous

I hope I answered this right...

7. jdoe0001

looks close

8. jdoe0001

from a polynomial, you'd get both, positive and negative roots so you find both, positive ones and negative ones then add up their possible combinations,a nd whatever is left, from the fundamental theorem, is a complex one

9. jdoe0001

let us take your example... one sec

10. jdoe0001

$$f(x)=7x^5-3x^2-9x+2\impliedby \begin{cases} \textit{2 changes}\\ \textit{positive ones} \\\hline\\ 2,0 \end{cases} \\ \quad \\ f(-x)=-7x^5-3x^2+9x+2\impliedby \begin{cases} \textit{1 change}\\ \textit{negative ones} \\\hline\\ 1 \end{cases}$$

11. jdoe0001

the polynomial degree is 5 so we have either 2 positive, 1 negative but the degree is 5, so the roots are also 5, so 2 are missing and those are our complex ones 2 positive, 1 negative, 2 complex 2 + 1 + 2 or 0 positive, 1 negative 5 roots, thus, 4 complex 0 positive, 1 negative, 4 complex 0 +1 + 4

12. jdoe0001

keeping in mind that, complex root, involve a radical solution, and thus usually come in pairs so, you'd end up getting 2 or 4 or 6 or 8 or so, but an "even" amount of complex

13. anonymous

Okay

14. jdoe0001

but as you see, we use the fundamental theorem, to get that there are 5 roots subtract the positive ones and negative ones, and what's left is complex