I know about Descartes Rule but I don't really get how to figure this out-Please check my answers-
I don't get this one find number of possible positive and negative real roots of f(x) = x^4-x^3+2x^2 + x-5
I think there are 3 sign changes so there are 3 positive but I get really confused on doing the negative real zeros would I rewrite for negative f(x) = -x^4 -(-x^3) -2x^2 -(x-5) so there would be 2negative real zeros
f(x) = x^4 - x^3 + 2x^2 + x - 5
has 3 sign changes.
So there would be either 3 or 1 positive roots.
f(-x) = (-x)^4 - (-x)^3 + 2(-x)^2 + (-x) - 5
= x^4 + x^3 + 2x^2 - x - 5
I see only one sign change, so there is 1 negative roots.
So there would be either 1+ root, 1- root, and 2 complex
or
there would be 3+ roots, and 1- root
remember we could have at most 4 roots
Looks this youtube clip showing the procedure.
http://www.youtube.com/watch?v=5YAmwfT3Esc
running it through "wolfram" shows the roots
http://www.wolframalpha.com/input/?i=x%5E4-x%5E3%2B2x%5E2+%2B+x-5+%3D0
show 1 positive, 1 negative, and 2 complex roots
Thank you-I viewed the youtube-that was really helpful and I did use Wolfram as you suggested. I even checked some of my other problems
To determine the number of possible positive and negative real roots of a polynomial using Descartes' Rule of Signs, you need to follow these steps:
1. Write down the polynomial in standard form, with descending powers of x: f(x) = x^4 - x^3 + 2x^2 + x - 5.
2. Count the number of sign changes in the coefficients of the polynomial. In your case, there are three sign changes (from positive to negative or vice versa): from +x^4 to -x^3, from -x^3 to +2x^2, and from +2x^2 to +x. Note that the constant term (-5) is not considered when counting sign changes.
3. The number of positive real roots of the polynomial is equal to the number of sign changes or is less than it by an even integer. In your case, there are three sign changes, so there are either 3 positive real roots or 1 positive real root (since 3 - 2 = 1).
4. To find the number of negative real roots, rewrite the polynomial by replacing each x with -x, giving you -f(x): f(-x) = (-x)^4 - (-x)^3 + 2(-x)^2 + (-x) - 5 = x^4 + x^3 + 2x^2 - x - 5.
5. Count the number of sign changes in this newly obtained polynomial, which is f(-x). In your case, there are two sign changes: from +x^4 to +x^3 and from +x to -5.
6. The number of negative real roots of the polynomial is equal to the number of sign changes or is less than it by an even integer. In your case, there are two sign changes, so there are either 2 negative real roots or 0 negative real roots (since 2 - 2 = 0).
Therefore, according to Descartes' Rule of Signs, your polynomial f(x) = x^4 - x^3 + 2x^2 + x - 5 has either 3 positive real roots or 1 positive real root and either 2 negative real roots or 0 negative real roots.