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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6. Determine whether x-1 is a factor of the polynomial P(x) = x^3 - 4x^2 + x + 6. Step 1: Apply the Factor Theorem. According to the Factor Theorem, (x-1) is a factor of P(x) if and only if P(1) = 0. Step 2: Evaluate P(1). P(1) = (1)^3 - 4(1)^2 + (1) + 6 P(1) = 1 - 4(1) + 1 + 6 P(1) = 1 - 4 + 1 + 6 P(1) = -3 + 1 + 6 P(1) = -2 + 6 P(1) = 4 Step 3: Conclude based on the result. Since P(1) = 4 ≠ 0, (x-1) is not a factor of P(x). The answer is No, x-1 is not a factor. 7. Given that x+2 is a factor of the polynomial P(x) = x^3 + x^2 - 4x - 4, find the complete factorization of P(x). Step 1: Use synthetic division to divide P(x) by (x+2). Since (x+2) is a factor, we divide by x = -2. ` -2 | 1 1 -4 -4 | -2 2 4 ------------------ 1 -1 -2 0 ` The quotient is x^2 - x - 2 and the remainder is 0. Step 2: Factor the quadratic quotient. The quadratic quotient is x^2 - x - 2. We need two numbers that multiply to -2 and add to -1. These numbers are -2 and 1. x^2 - x - 2 = (x - 2)(x + 1) Step 3: Write the complete factorization of P(x). P(x) = (x+2)(x^2 - x - 2) P(x) = (x+2)(x-2)(x+1) The complete factorization is P(x) = (x+2)(x-2)(x+1). 8. Explain why the following expression is not a polynomial: f(x) = (3)/(x^2) + x. A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The term (3)/(x^2) can be written as 3x^-2. Since the exponent of x in this term is -2, which is a negative integer, the expression f(x) does not meet the definition of a polynomial. 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, the remainder when P(x) is divided by (x+1) is P(-1). Step 2: Evaluate P(-1). P(-1) = 3(-1)^4 - 2(-1)^3 + 5(-1) - 7 P(-1) = 3(1) - 2(-1) - 5 - 7 P(-1) = 3 + 2 - 5 - 7 P(-1) = 5 - 5 - 7 P(-1) = 0 - 7 P(-1) = -7 The remainder is -7. 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. For P(1)=6: (1)^3 + a(1)^2 + b(1) + c = 6 1 + a + b + c = 6 (1) a + b + c = 5 For P(2)=15: (2)^3 + a(2)^2 + b(2) + c = 15 8 + 4a + 2b + c = 15 (2) 4a + 2b + c = 7 For P(3)=28: (3)^3 + a(3)^2 + b(3) + c = 28 27 + 9a + 3b + c = 28 (3) 9a + 3b + c = 1 Step 2: Solve the system of linear equations. Subtract (1) from (2): (4a + 2b + c) - (a + b + c) = 7 - 5 (4) 3a + b = 2 Subtract (2) from (3): (9a + 3b + c) - (4a + 2b + c) = 1 - 7 (5) 5a + b = -6 Subtract (4) from (5): (5a + b) - (3a + b) = -6 - 2 2a = -8 a = -4 Substitute a = -4 into (4): 3(-4) + b = 2 -12 + b = 2 b = 14 Substitute a = -4 and b = 14 into (1): -4 + 14 + c = 5 10 + c = 5 c = -5 Step 3: Determine the polynomial P(x). Substitute the values a = -4, b = 14, and c = -5 into P(x) = x^3 + ax^2 + bx + c. P(x) = x^3 - 4x^2 + 14x - 5 The values are a = -4, b = 14, c = -5. The polynomial is P(x) = x^3 - 4x^2 + 14x - 5.