Use the Remainder Theorem for the first condition.

Step 1: Use the Remainder Theorem for the first condition. The polynomial is f(x) = 3 - ax + bx^2 - x^3. When f(x) is divided by (x+2), the remainder is 45. By the Remainder Theorem, f(-2) = 45. Substitute x = -2 into f(x): f(-2) = 3 - a(-2) + b(-2)^2 - (-2)^3 45 = 3 + 2a + 4b - (-8) 45 = 3 + 2a + 4b + 8 45 = 11 + 2a + 4b 34 = 2a + 4b Divide the equation by 2: 17 = a + 2b (*) Step 2: Use the Factor Theorem for the second condition. (3-x) is a factor of f(x). By the Factor Theorem, if (3-x) is a factor, then f(3) = 0. Substitute x = 3 into f(x): f(3) = 3 - a(3) + b(3)^2 - (3)^3 0 = 3 - 3a + 9b - 27 0 = -24 - 3a + 9b Rearrange the equation: 24 = -3a + 9b Divide the equation by 3: 8 = -a + 3b (**) Step 3: Solve the system of linear equations for a and b. We have two equations: 1) a + 2b = 17 2) -a + 3b = 8 Add equation (1) and equation (2): (a + 2b) + (-a + 3b) = 17 + 8 5b = 25 b = (25)/(5) b = 5 Substitute b=5 into equation (1): a + 2(5) = 17 a + 10 = 17 a = 17 - 10 a = 7 The values of the real constants are a=7 and b=5. a=7, b=5 ii) Express (5x)/((x+3)(x-2)) into partial fractions. Step 1: Set up the partial fraction decomposition. Since the denominator has two distinct linear factors, (x+3) and (x-2), we can write: (5x)/((x+3)(x-2)) = (A)/(x+3) + (B)/(x-2) Step 2: Clear the denominators. Multiply both sides by (x+3)(x-2): 5x = A(x-2) + B(x+3) Step 3: Solve for A and B using substitution. To find A, set x = -3: 5(-3) = A(-3-2) + B(-3+3) -15 = A(-5) + B(0) -15 = -5A A = (-15)/(-5) A = 3 To find B, set x = 2: 5(2) = A(2-2) + B(2+3) 10 = A(0) + B(5) 10 = 5B B = (10)/(5) B = 2 Step 4: Write the partial fraction decomposition. Substitute the values of A and B back into the decomposition: (5x)/((x+3)(x-2)) = (3)/(x+3) + (2)/(x-2) (3)/(x+3) + (2)/(x-2) Send me the next one 📸