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.
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You're on a roll — To find the values of m and n, 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 factor x^2 + x - 2. We factorize the quadratic expression: x^2 + x - 2 = 0 (x+2)(x-1) = 0 The roots are x = -2 and x = 1. Step 2: Apply the Factor Theorem. Since x=-2 and x=1 are roots, we must have f(-2) = 0 and f(1) = 0. The polynomial is f(x) = 2x^3 + 9x^2 + mx - n. For x=1: f(1) = 2(1)^3 + 9(1)^2 + m(1) - n = 0 2 + 9 + m - n = 0 11 + m - n = 0 m - n = -11 (Equation 1) For x=-2: f(-2) = 2(-2)^3 + 9(-2)^2 + m(-2) - n = 0 2(-8) + 9(4) - 2m - n = 0 -16 + 36 - 2m - n = 0 20 - 2m - n = 0 -2m - n = -20 2m + n = 20 (Equation 2) Step 3: Solve the system of linear equations for m and n. We have the system: 1) m - n = -11 2) 2m + n = 20 Add Equation 1 and Equation 2: (m - n) + (2m + n) = -11 + 20 3m = 9 m = (9)/(3) m = 3 Substitute the value of m=3 into Equation 1: 3 - n = -11 -n = -11 - 3 -n = -14 n = 14 The values are m=3 and n=14. The final answer is m=3, n=14. 4 down, 1 left today. What's next?