This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To solve Question 1.3, we will use the Factor Theorem. If (x^2 + x - 2) is a factor of f(x), then the roots of (x^2 + x - 2) = 0 must also be roots of f(x) = 0. Step 1: Find the roots of the quadratic factor. Factorize the quadratic expression x^2 + x - 2: x^2 + x - 2 = 0 (x+2)(x-1) = 0 The roots are x = -2 and x = 1. Step 2: Use the Factor Theorem to set up equations. Since (x+2) and (x-1) are factors of f(x) = x^3 + px^2 + 2x + q, we know that f(-2) = 0 and f(1) = 0. For x=1: f(1) = (1)^3 + p(1)^2 + 2(1) + q = 0 1 + p + 2 + q = 0 p + q + 3 = 0 p + q = -3 (Equation 1) For x=-2: f(-2) = (-2)^3 + p(-2)^2 + 2(-2) + q = 0 -8 + 4p - 4 + q = 0 4p + q - 12 = 0 4p + q = 12 (Equation 2) Step 3: Solve the system of linear equations for p and q. We have the system: 1) p + q = -3 2) 4p + q = 12 Subtract Equation 1 from Equation 2: (4p + q) - (p + q) = 12 - (-3) 4p - p + q - q = 12 + 3 3p = 15 p = (15)/(3) p = 5 Step 4: Substitute the value of p into Equation 1 to find q. p + q = -3 5 + q = -3 q = -3 - 5 q = -8 The values are p=5, q=-8. 3 done, 2 left today. You're making progress.