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
1) (b) Find Q(x). Step 1: Determine the degree of Q(x). The given polynomial is P(x) = (2x-1)(x-3)Q(x) + 12x - 8. The term (2x-1)(x-3) is a quadratic, which has a degree of 2. We are given that P(x) is of degree 3. For the product (2x-1)(x-3)Q(x) to result in a polynomial of degree 3 (after considering the highest degree term), Q(x) must be a linear polynomial (degree 1). Let Q(x) = ax+b. Step 2: Use the condition that P(x) is exactly divisible by x-1. By the Factor Theorem, if P(x) is exactly divisible by x-1, then P(1) = 0. Substitute x=1 into the expression for P(x): P(1) = (2(1)-1)(1-3)Q(1) + 12(1) - 8 P(1) = (1)(-2)Q(1) + 12 - 8 P(1) = -2Q(1) + 4 Since P(1) = 0: 0 = -2Q(1) + 4 2Q(1) = 4 Q(1) = 2 Step 3: Use the condition that P(0) = 10. Substitute x=0 into the expression for P(x): P(0) = (2(0)-1)(0-3)Q(0) + 12(0) - 8 P(0) = (-1)(-3)Q(0) + 0 - 8 P(0) = 3Q(0) - 8 Since P(0) = 10: 10 = 3Q(0) - 8 18 = 3Q(0) Q(0) = 6 Step 4: Find Q(x) using the values of Q(0) and Q(1). We have Q(x) = ax+b. Using Q(0) = 6: a(0) + b = 6 b = 6 So, Q(x) = ax+6. Using Q(1) = 2: a(1) + 6 = 2 a + 6 = 2 a = 2 - 6 a = -4 Therefore, Q(x) = -4x+6. The final answer is -4x+6.