Given f(x) = 3 - ax + bx2 - x3, find the values of a and b if f(x) divided by (x+2) gives a remainder of 45 and (3-x) is a factor of f(x). Also decompose (5x)/((x+3)(x-2)) into partial fractions.

Hey Dany — ready when you are. 1. (i) Step 1: Use the Remainder Theorem for the first condition. Given f(x) = 3 - ax + bx^2 - x^3. When f(x) is divided by (x+2), the remainder is 45. This means f(-2) = 45. f(-2) = 3 - a(-2) + b(-2)^2 - (-2)^3 = 45 3 + 2a + 4b - (-8) = 45 3 + 2a + 4b + 8 = 45 11 + 2a + 4b = 45 2a + 4b = 34 Divide by 2: a + 2b = 17 (Equation 1) Step 2: Use the Factor Theorem for the second condition. Given that (3-x) is a factor of f(x). This means f(3) = 0. f(3) = 3 - a(3) + b(3)^2 - (3)^3 = 0 3 - 3a + 9b - 27 = 0 -24 - 3a + 9b = 0 -3a + 9b = 24 Divide by 3: -a + 3b = 8 (Equation 2) Step 3: Solve the system of linear equations. Add Equation 1 and Equation 2: (a + 2b) + (-a + 3b) = 17 + 8 5b = 25 b = 5 Substitute b=5 into Equation 1: a + 2(5) = 17 a + 10 = 17 a = 7 The values of the real constants are a=7, b=5. 1. (ii) Step 1: Set up the partial fraction decomposition. (5x)/((x+3)(x-2)) = (A)/(x+3) + (B)/(x-2) Multiply both sides by (x+3)(x-2): 5x = A(x-2) + B(x+3) Step 2: Solve for A by substituting x=-3. 5(-3) = A(-3-2) + B(-3+3) -15 = A(-5) + B(0) -15 = -5A A = 3 Step 3: Solve for B by substituting x=2. 5(2) = A(2-2) + B(2+3) 10 = A(0) + B(5) 10 = 5B B = 2 Step 4: Write the partial fraction decomposition. (5x)/((x+3)(x-2)) = (3)/(x+3) + (2)/(x-2) Send me the next one 📸