What does Descartes' Rule of Signs say about the number of positive real roots and negative real roots for the polynomial function
p(x)=-x^3+4x^2-6x+3
What are the possible numbers of positive real roots?
What are the possible numbers of negative real roots?
Descartes' Rule of Signs states that for a polynomial function with real coefficients, the number of positive real roots is equal to the number of sign changes in the coefficients or fewer by an even integer, while the number of negative real roots is equal to the number of sign changes in the coefficients of the function f(-x) or fewer by an even integer.
For the given polynomial function p(x) = -x^3 + 4x^2 - 6x + 3, we can determine the number of sign changes in the coefficients.
The coefficients of the polynomial function are -1, 4, -6, and 3.
Number of sign changes in the coefficients: 3
According to Descartes' Rule of Signs, the possible numbers of positive real roots are either 3 or 1 (three sign changes or one sign change by an even integer).
To find the possible number of negative real roots, we substitute -x in place of x and evaluate the function f(-x).
f(-x) = -(-x)^3 + 4(-x)^2 - 6(-x) + 3
= -(-x^3) + 4x^2 + 6x + 3
= x^3 + 4x^2 + 6x + 3
We can now determine the number of sign changes in the coefficients of this new function.
The coefficients of the new function are 1, 4, 6, and 3.
Number of sign changes in the coefficients: 2
According to Descartes' Rule of Signs, the possible numbers of negative real roots are either 2 or 0 (two sign changes or zero sign changes by an even integer).
Therefore, the possible numbers of positive real roots for the polynomial function p(x) = -x^3 + 4x^2 - 6x + 3 are either 3 or 1.
The possible numbers of negative real roots are either 2 or 0.