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Part 1
Use Descartes' rule of signs to determine the possible number of positive real roots and the possible number of negative real roots for the following function.
P(x)=x^3-2x^2-5x+6
What is the possible number of positive real roots of this function?
4 or 2
3 or 1
2 or 1
2 or 0
What is the possible number of negative real roots of this function?
1
3 or 2
2 or 0
3 or 1
To determine the possible number of positive real roots, count the number of sign changes in the function P(x). In this case, we have two sign changes: from positive to negative between the terms x^3 and -2x^2, and from negative to positive between the terms -5x and 6.
According to Descartes' rule of signs, the possible number of positive real roots of a polynomial is either equal to the number of sign changes or a number less than that by an even whole number. In this case, the possible number of positive real roots is either 2 or 0.
To determine the possible number of negative real roots, count the number of sign changes in the function P(-x). In this case, we have one sign change: from positive to negative between the terms (-x)^3 and -2(-x)^2.
According to Descartes' rule of signs, the possible number of negative real roots of a polynomial is either equal to the number of sign changes or a number less than that by an even whole number. In this case, the possible number of negative real roots is either 1 or 0.
Therefore, the possible number of positive real roots of P(x) is either 2 or 0, and the possible number of negative real roots is either 1 or 0. So, the correct answer options are:
Possible number of positive real roots: 2 or 0
Possible number of negative real roots: 1 or 0