Question

How many and of which kind of roots does the equation f(x)=x^3+2x^2+4x+8 have?

Answer 1

The cubic equation f(x) = x³ + 2 · x² + 4 · x + 8 has one real root and two complex roots.

What kind of roots does have a cubic equation?

In this problem we have a cubic equation and the nature of their roots must be inferred according to a algebraic method.

Cubic equations are polynomials of the form y = a · x³ + b · x² + c · x + d, there is a method to infer the nature of the roots of such polynomials: The discriminant from Cardano's method, an analytical method used to solve polynomials of the form a · x³ + b · x² + c · x + d = 0.

The discriminant is described below:

Δ = 18 · a · b · c · d - 4 · b³ · d + b² · c² - 4 · a · c³ - 27 · a² · d²      (1)

Where:

1. There are three distinct real roots for Δ > 0.
2. Real roots with multiplicity greater than 1 for Δ = 0.
3. A real root and two complex roots for Δ < 0.

If we know that a = 1, b = 2, c = 4 and d = 8, then the nature of the roots is:

Δ = 18 · 1 · 2 · 4 · 8 - 4 · 2³ · 8 + 2² · 4² - 4 · 1 · 4³ - 27 · 1² · 8²

Δ = - 1024

The cubic equation f(x) = x³ + 2 · x² + 4 · x + 8 has one real root and two complex roots.

To learn more on cubic equations: https://brainly.com/question/13730904

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