Determine what Descartes Rule of Signs says about the number of positive and negative real roots for the polynomial function
P(x) = 9x^3 - 4x^2 +10
Descartes' Rule of Signs states that the number of positive real roots of a polynomial function is equal to the number of sign changes in its coefficients or fewer by an even integer. Similarly, the number of negative real roots is equal to the number of sign changes in the coefficients of P(-x) or fewer by an even integer.
For the given function P(x) = 9x^3 - 4x^2 + 10, we need to determine the number of sign changes in the coefficients.
The coefficients are:
a = 9 (coefficient of x^3)
b = -4 (coefficient of x^2)
c = 0 (coefficient of x) since there is no x-term in the function
d = 10 (constant term)
There is a sign change between a and b, and no sign change between b and c since c is 0. There is also a sign change between c and d. Therefore, there are 2 sign changes in the coefficients.
According to Descartes' Rule of Signs, the number of positive real roots of P(x) can either be 2 or 0. It cannot be any odd number.
To find the number of negative real roots, we substitute -x for x in the given function and observe the sign changes in the coefficients:
P(-x) = 9(-x)^3 - 4(-x)^2 + 10 = -9x^3 - 4x^2 + 10
Again, there are 2 sign changes between the coefficients of P(-x). Hence, the number of negative real roots can either be 2 or 0.
In summary, Descartes' Rule of Signs tells us that the number of positive real roots of the function P(x) = 9x^3 - 4x^2 + 10 can be either 2 or 0, and the number of negative real roots can also be either 2 or 0.