2. Consider the real polynomial: q (x) = x3 — 2x2 + ax — b + 1 Determine the values of a and b so that q (x) + 1 is divisible by (x + 2) and q (x) has a root at x = 1.

To determine the values of a and b, we'll use the given conditions: q(x) + 1 is divisible by (x + 2). q(x) has a root at x = 1. Let's address these conditions one by one. Condition 1: q(x) + 1 is divisible by (x + 2) For a polynomial to be divisible by (x + 2), it must have (x + 2) as a factor. Therefore, we can write q(x) + 1 as (x + 2) multiplied by some other polynomial, say p(x): q(x) + 1 = (x + 2) * p(x) Expanding q(x) and simplifying: x^3 - 2x^2 + ax - b + 1 = xp(x) + 2p(x) Comparing the coefficients on both sides, we have: x^3 = xp(x) (coefficients of x^3 on both sides) -2x^2 + ax - b + 1 = 2p(x) (coefficients of x^2, x, and constant term on both sides) From equation 1), we see that p(x) must be x^2 for x^3 to be equal to xp(x). Substituting p(x) = x^2 into equation 2): -2x^2 + ax - b + 1 = 2x^2 Now, let's simplify this equation further. 3x^2 + ax - (b + 1) = 0 Now, we have a quadratic equation. To satisfy condition 1, the discriminant of this quadratic equation must be zero, since (x + 2) should be a factor of q(x) + 1. The discriminant is given by: D = a^2 - 4ac Substituting the coefficients: D = a^2 - 4 * 3 * (-(b + 1)) D = a^2 + 12b + 12 For the discriminant to be zero: a^2 + 12b + 12 = 0 ----(Equation 3) Now, let's move on to condition 2. Condition 2: q(x) has a root at x = 1 For q(x) to have a root at x = 1, we substitute x = 1 into the polynomial q(x) and equate it to zero: q(1) = 1^3 - 2(1)^2 + a(1) - b + 1 = 0 Simplifying this equation: 1 - 2 + a - b + 1 = 0 a - b = 0 ----(Equation 4) Now, we have two equations (Equation 3 and Equation 4) involving the variables a and b. We can solve these simultaneous equations to find the values of a and b. From Equation 4, we have: a = b Substituting this value of a into Equation 3: b^2 + 12b + 12 = 0 This is a quadratic equation in b. Solving it using any quadratic equation solver, we find the values of b: b = -6 ± √(6) Since a = b, the corresponding values of a are: a = -6 ± √(6) Therefore, the values of a and b that satisfy the given conditions are: a = -6 + √(6) b = -6 + √(6) or a = -6 - √(6) b = -6 - √(6)