Which correctly describes the roots of the following cubic equation x³ + 6x² + 11x + 6
One real root, two complex roots
Two real roots and one complex root
Three real roots, two of which are equal value
Three real roots, each with a different value

Solution:

The given equation f(x) = x³ + 6x² + 11x + 6 is a polynomial of degree three and hence it has three roots.

Rewriting the given equation we get:

x³ + 5x² + x² + 5x + 6x + 6

x³ + 5x² + 6x + x² + 5x + 6

x(x² + 5x + 6) + x² + 5x + 6

Taking x² + 5x + 6 as common we get

(x + 1)(x² + 5x + 6)

(x + 1)(x² + 3x + 2x + 6)

(x + 1)(x(x + 3) + 2(x + 3))

(x + 1)(x + 2)(x + 3)

Hence we get three factors of the given equation from which also give the three zeroes of the function.

x + 1 = 0 ⇒ x = -1      The first root

x + 2 = 0 ⇒ x = -2      The second root

x + 3 = 0 ⇒ x = -3      The third root

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Which correctly describes the roots of the following cubic equation x³ + 6x² + 11x + 6

Summary:

Three real roots, each with a different value correctly describes the roots of the following cubic equation x³ + 6x² + 11x + 6