Which correctly describes the roots of the following cubic equation x^3+6x^2+11x+ a. one real root, two complex roots b. two real roots and one complex root c. three real roots, two of which are equal value d. three real roots, each with a different value

Option D is correct that is Three real roots, each with a different value.

Step-by-step explanation:

Given Cubic Equation,

x³ + 6x² + 11x + 6

We have to find nature of roots of the equation.

We Answer by finding roots of the equation.

Let p(x) = x³ + 6x² + 11x + 6

put x = -1

p(-1) = (-1)³ + 6(-1)² + 11(-1) + 6 = -1 + 6 - 11 + 6 = 0

So, x = -1   is first zero and ( x + 1 )  is first factor.

To find other zeroes we divide p(x) by ( x + 1 )

On division, we get quotient = x² + 5x + 6

Zeroes of obtained quadratic polynomial are roots of p(x)

x² + 5x + 6 = 0

x² + 2x + 3x + 6 = 0

x( x + 2 ) + 3 ( x + 2 ) = 0

( x + 2 ) ( x + 3 ) = 0

x = -2 & -3

Thus, Roots of given cubic equation are -1 , -2 , -3

Therefore, Option D is correct that is Three real roots, each with a different value.

Pic is attached.