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.

Evaluating f(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.

The polynomial can be factored as (x - 1)(x - 2)(x - 3), so x - 1 is one of its factors.

Substituting x = 1 into the polynomial gives f(1) = 1 - 4 + 6 - 4 + 1 = 0.

The polynomial can be rewritten in a form that shows symmetry about the line x = 1.

A polynomial has a double root when its discriminant is equal to zero.

Using Vieta's formulas, the sum of the roots (-2 + 3) = 1, which means b = -1.

Using Vieta's formulas, the sum of the roots (3 + (-2)) = 1, hence b = -1.

According to Vieta's formulas, the sum of the roots r1 + r2 of the polynomial x^2 - 5x + 6 is equal to the coefficient of x (which is -(-5)) = 5.

Factoring the polynomial gives (x - 2)(x - 3), so the roots are 2 and 3.

Factoring the polynomial P(x) gives (x - 2)(x - 3), so the roots are 2 and 3.

Calculating P(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.

P(2) = 2^3 - 6(2^2) + 11(2) - 6 = 8 - 24 + 22 - 6 = 0.

Population after 1 year = 1000 * 1.10 = 1100. Population after 2 years = 1100 * 1.10 = 1210.

Population after one year = Current Population * (1 + Growth Rate) = 1 million * (1 + 0.02) = 1 million * 1.02 = 1.02 million.

Population after 2 years = 100,000 * (1 + 0.05)^2 = 100,000 * 1.1025 = 110,250.

Population after 1 year = 1000 * 1.10 = 1100. Population after 2 years = 1100 * 1.10 = 1210.

Distance = √(5^2 + 12^2) = √(25 + 144) = √169 = 13

Velocity vector = dr/dt = (2, 3, 4)

Distance = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29 ≈ 5.385.

Projection of P onto a = ((P · a) / |a|^2) * a.

Substituting t = 1, r = (2*1, 3*1, 4*1) = (2, 3, 4).

Vector PQ = Q - P = (1 - 3, 4 - (-2)) = (-2, 6).

Vector PQ = Q - P = (1 - 3, 2 - 4) = (-2, -2).

Distance = |r| = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29.

The x-coordinate of point P is the coefficient of i in the position vector, which is 2.

The x-component of r is 2.

Distance = |r| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Distance (d) = Potential (V) / Electric field (E) = 5 V / 2 V/m = 2.5 m.

The energy stored in a capacitor is given by U = 1/2 CV^2. If V is halved, the energy becomes U' = 1/2 C(V/2)^2 = 1/8 CV^2, which is halved.