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? |
Descartes rule is really helpful because it eliminates the long list of possible rational roots and you can tell how many positives or negatives roots you will have. Fundamental Theorem of Algebra finds the maximum number of zeros which includes real and complex numbers.
f(x)= x^3 – 6x^2 + 13x – 20
Yes Yes Yes ( the sign changes from one to the other)
so now we can have 3 or 1 positives.
now to find the negative
f(-x)= -x^3 - 6x^2 – 13x -20
No No No ( the signs did not change at all)
therefore, we have 0 negatives.
Then whenever you make your list, you can eliminate all of the negatives since it would be included in your answer. Since the degree is three, we are going to have three zeros. Positive | 3 | 1 |
Negative | 0 | 0 |
Imaginary | 0 | 2 |
In this example, your answer could have all three positives number or one positive number and two imaginary.
Now let's try another!
f(x)= -x^4 – 2x^3 – 8x + 2
No No Yes (one positive)
f(-x)= -x^4 + 2x^3 + 8x +2
Yes No No (one negative)
Positive | 1 |
Negative | 1 |
Imaginary | 2 |
In this example,the only roots you can have is one positive and one negative the other two are imaginary since the degree of this polynomial is four.