Question

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with real coefficients. 1....

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with real coefficients.

1. Deduce that either f(x) factors in R[x] as the product of three degree-one
polynomials, or f(x) factors in R[x] as the product of a degree-one
polynomial and an irreducible degree-two polynomial.

2.Deduce that either f(x) has three real roots (counting multiplicities) or
f(x) has one real root and two non-real (complex) roots that are complex
conjugates of each other.

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