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-c) is a factor of a polynomial P(x) if and only if P(c) = 0. In this case, c=1. 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). 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 use the root x=-2 for synthetic division. c|cccc -2 & 1 & 1 & -4 & -4 \\ & & -2 & 2 & 4 \\ & 1 & -1 & -2 & 0 \\ The remainder is 0, confirming that (x+2) is a factor. The quotient is x^2 - x - 2. Step 2: Factor the quadratic quotient. We need to factor x^2 - x - 2. We look for 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) 8. Explain why the following expression is not a polynomial: f(x) = (3)/(x^2) + x. A polynomial is an expression where all variables have non-negative integer exponents. The given expression can be rewritten as f(x) = 3x^-2 + x^1. The term 3x^-2 has an exponent of -2 for the variable x. Since -2 is a negative integer, it violates the definition of a polynomial.