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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3. If (x-5) is a factor of x^3 - 4x^2 - 11x + 30, find the remaining factors. Step 1: Since (x-5) is a factor, we can use synthetic division with x=5 to divide the polynomial x^3 - 4x^2 - 11x + 30. c|cccc 5 & 1 & -4 & -11 & 30 \\ & & 5 & 5 & -30 \\ & 1 & 1 & -6 & 0 \\ The coefficients of the quotient are 1, 1, -6, and the remainder is 0. This means the quotient is x^2 + x - 6. Step 2: Factor the quadratic expression obtained from the division. We need to factor x^2 + x - 6. We look for two numbers that multiply to -6 and add to 1. These numbers are 3 and -2. So, the quadratic factors as: x^2 + x - 6 = (x+3)(x-2) Step 3: Identify the remaining factors. The remaining factors are (x+3) and (x-2). Comparing this with the given options: A. (x-3) and (x+2) B. (x-3) and (x-2) C. (x+3) and (x-2) D. (x+3) and (x+2) The correct option is C. The final answer is C