By substituting x = 1 into the equation, we find that it satisfies the equation, hence 1 is a root.

The roots can be found using the Rational Root Theorem and synthetic division, confirming that all roots are real.

The discriminant is 2^2 - 4(1)(1) = 0, indicating that the roots are real and equal.

The discriminant must be non-negative: 4^2 - 4*1*k >= 0 leads to 16 - 4k >= 0, thus k <= 4.

The condition for no real roots is that the discriminant must be less than zero: 6^2 - 4*1*k < 0 => 36 < 4k => k > 9.

The equation can be factored as (x-1)^3 = 0, which has one real root (x = 1) with multiplicity 3.

By substituting x = 2 into the equation, we find that it satisfies the equation, thus x = 2 is a root.

The product of the roots of the cubic equation ax^3 + bx^2 + cx + d = 0 is given by -d/a. Here, d = -6 and a = 1, so the product is -(-6)/1 = 6.

By substituting x = 2 into the equation, we find that 2 is a root since 2^3 - 6(2^2) + 11(2) - 6 = 0.

The polynomial can be factored as (x-1)^3, indicating that it has one real root with multiplicity 3, hence all roots are real and equal.

The sum of the roots is given by -b/a = 3/1 = 3.

The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.

The product of the roots is given by c/a. Here, c = 6 and a = 1, so αβ = 6.

The discriminant must be negative for no real roots: 5^2 - 4*1*k < 0, which simplifies to 25 - 4k < 0, or k > 25.

For both roots to be negative, the sum of the roots (which is -5) must be negative, and the product (k) must be positive. Thus, k > 0.

By Vieta's formulas, the sum of the roots (a + b + c) of the polynomial x^3 - 6x^2 + 11x - 6 is equal to the coefficient of x^2 with the opposite sign, which is 6.

The discriminant is given by b^2 - 4ac. Here, b = 2, a = 1, c = 1, so the discriminant is 2^2 - 4*1*1 = 0.

The discriminant must be positive: 3^2 - 4*1*k > 0 leads to 9 - 4k > 0, thus k < 9.

For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 leads to k = 4.

The product of the roots ab is given by c/a. Here, c = 6 and a = 1, so ab = 6.

The product of the roots is (-2)(-3) = 6, so k = 6.

For real and distinct roots, the discriminant must be greater than zero: 6^2 - 4*1*k > 0 leads to k < 9.

Using Vieta's formulas, n = 3 * 4 = 12.

The discriminant must be positive for real and distinct roots: (-2)^2 - 4*1*k > 0, which simplifies to 4 - 4k > 0, or k < 1.

The discriminant must be greater than zero for real and distinct roots: (-4)^2 - 4*1*k > 0, which simplifies to 16 - 4k > 0, or k < 4.

By Vieta's formulas, ab = 10, which is the constant term of the polynomial.

By Vieta's formulas, the sum of the roots a + b + c = -(-3) = 3.

For the roots to be equal, the discriminant must be zero. Thus, 2^2 - 4*1*k = 0 leads to k = 1.

For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4(1)(k) = 0 leads to k = 4.

The discriminant is 0, which indicates that the roots are real and equal.