For the equation, state the number of complex roots, the possible number of real roots, and the possible rational roots. x^3 − 6x^2 + 11x − 6 = 0

Answer 1

1) The number of complex roots is 3 since the degree of the polynomial is 3.

2)The number of complex roots (including multiplicity) of a polynomial function is equal to the degree of the polynomial functions.

The possible number of real roots is either 1 or 3.

Since the imaginary roots of a polynomial function with real coefficients come as conjugate pairs, if there are 2 imaginary roots, then there is 1 real root; if there are no imaginary roots, then there are 3 real roots.

The possible number of rational roots is either 1 or 3.

We can use the synthetic division to find the first rational root (assume it exists). We try 1 first since it's the smallest positive integer. As we can see it works. So 1 is a root of the polynomial function.

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