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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Here are the solutions for problems 9 and 10. Problem 9: Find the remainder when P(x) = 3x^4 - 2x^3 + 5x - 7 is divided by x+1. Step 1: Apply the Remainder Theorem. According to the Remainder Theorem, if a polynomial P(x) is divided by (x-a), the remainder is P(a). Here, the divisor is x+1, so a=-1. The remainder is P(-1). Step 2: Substitute x=-1 into P(x). P(-1) = 3(-1)^4 - 2(-1)^3 + 5(-1) - 7 P(-1) = 3(1) - 2(-1) + 5(-1) - 7 P(-1) = 3 + 2 - 5 - 7 P(-1) = 5 - 5 - 7 P(-1) = 0 - 7 P(-1) = -7 The remainder is -7. Problem 10: Let P(x) = x^3 + ax^2 + bx + c be a polynomial with real coefficients. Suppose that P(1)=6, P(2)=15, P(3)=28. Find the values of a, b, and c. Hence determine the polynomial P(x). Step 1: Set up a system of equations using the given conditions. Given P(x) = x^3 + ax^2 + bx + c: For P(1)=6: 1^3 + a(1)^2 + b(1) + c = 6 1 + a + b + c = 6 a + b + c = 5 (1) For P(2)=15: 2^3 + a(2)^2 + b(2) + c = 15 8 + 4a + 2b + c = 15 4a + 2b + c = 7 (2) For P(3)=28: 3^3 + a(3)^2 + b(3) + c = 28 27 + 9a + 3b + c = 28 9a + 3b + c = 1 (3) Step 2: Solve the system of equations. Subtract (1) from (2): (4a + 2b + c) - (a + b + c) = 7 - 5 3a + b = 2 (4) Subtract (2) from (3): (9a + 3b + c) - (4a + 2b + c) = 1 - 7 5a + b = -6 (5) Subtract (4) from (5): (5a + b) - (3a + b) = -6 - 2 2a = -8 a = -4 Substitute a=-4 into equation (4): 3(-4) + b = 2 -12 + b = 2 b = 14 Substitute a=-4 and b=14 into equation (1): -4 + 14 + c = 5 10 + c = 5 c = -5 Step 3: State the values of a, b, c and the polynomial P(x). The values are a = -4, b = 14, and c = -5. The polynomial P(x) is x^3 - 4x^2 + 14x - 5. The final answer is a=-4, b=14, c=-5, P(x) = x^3 - 4x^2 + 14x - 5.