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
Step 1: Use the given information about P(x). The polynomial is given by P(x) = (2x-1)(x-3)Q(x) + 12x-8. Step 2: Use the condition that P(x) is exactly divisible by x-1. According to 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 = 0 (1)(-2)Q(1) + 12-8 = 0 -2Q(1) + 4 = 0 -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 = 10 (-1)(-3)Q(0) + 0-8 = 10 3Q(0) - 8 = 10 3Q(0) = 18 Q(0) = 6 Step 4: Determine the degree of Q(x). The polynomial P(x) has a degree of 3. The term (2x-1)(x-3) is a quadratic expression (degree 2). The term 12x-8 is a linear expression (degree 1). For P(x) to be of degree 3, the product (2x-1)(x-3)Q(x) must be of degree 3. Since (2x-1)(x-3) is degree 2, Q(x) must be a linear polynomial (degree 1). Let Q(x) = ax+b. Step 5: Use the values of Q(1) and Q(0) to find a and b. From Q(1)=2: a(1)+b = 2 a+b = 2 (*) From Q(0)=6: a(0)+b = 6 b = 6 Substitute b=6 into equation (*): a+6 = 2 a = 2-6 a = -4 Step 6: Write the expression for Q(x). Using a=-4 and b=6: Q(x) = -4x+6 The final answer is Q(x) = -4x+6.