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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1) a) Let the given expression be P(x) = bx^3 - x^2 + cx - b. Since (x+1) is a factor, P(-1) = 0. b(-1)^3 - (-1)^2 + c(-1) - b = 0 -b - 1 - c - b = 0 -2b - c - 1 = 0 2b + c = -1 (1) Since (x-2) is a factor, P(2) = 0. b(2)^3 - (2)^2 + c(2) - b = 0 8b - 4 + 2c - b = 0 7b + 2c - 4 = 0 7b + 2c = 4 (2) From equation (1), c = -1 - 2b. Substitute this into equation (2): 7b + 2(-1 - 2b) = 4 7b - 2 - 4b = 4 3b - 2 = 4 3b = 6 b = 2 Now substitute b=2 back into c = -1 - 2b: c = -1 - 2(2) c = -1 - 4 c = -5 The values are b=2, c=-5. b) Substitute the values of b and c into the expression: P(x) = 2x^3 - x^2 - 5x - 2 The expression is 2x^3 - x^2 - 5x - 2. c) To show that (x-4) is not a factor, we evaluate P(4). If P(4) ≠ 0, then (x-4) is not a factor. P(4) = 2(4)^3 - (4)^2 - 5(4) - 2 P(4) = 2(64) - 16 - 20 - 2 P(4) = 128 - 16 - 20 - 2 P(4) = 112 - 20 - 2 P(4) = 92 - 2 P(4) = 90 Since P(4) = 90 ≠ 0, (x-4) is not a factor of the expression. d) We know that (x+1) and (x-2) are factors. Their product is: (x+1)(x-2) = x^2 - 2x + x - 2 = x^2 - x - 2 Now, we divide P(x) = 2x^3 - x^2 - 5x - 2 by (x^2 - x - 2) to find the remaining factor. Using polynomial long division: c|cc cc 2r2x & +1 \\ 2-5 x^2-x-2 & 2x^3 & -x^2 & -5x & -2 \\ 2r-(2x^3 & -2x^2 & -4x) \\ 2-4 2r0 & x^2 & -x & -2 \\ 2r & -(x^2 & -x & -2) \\ 3-5 2r & 0 & 0 & 0 \\ The remaining factor is (2x+1). To solve for x when the expression is equated to zero: (x+1)(x-2)(2x+1) = 0 Setting each factor to zero: x+1 = 0 x = -1 x-2 = 0 x = 2 2x+1 = 0 2x = -1 x = -(1)/(2) The remaining factor is 2x+1 and the values of x are -1, 2, -(1)/(2). 2) Express f(x) = (x)/(x^2 - 4) in partial fractions. First, factor the denominator: x^2 - 4 = (x-2)(x+2). So, f(x) = (x)/((x-2)(x+2)). We set up the partial fraction decomposition: (x)/((x-2)(x+2)) = (A)/(x-2) + (B)/(x+2) Multiply both sides by (x-2)(x+2): x = A(x+2) + B(x-2) To find A, let x=2: 2 = A(2+2) + B(2-2) 2 = 4A + 0 4A = 2 A = (2)/(4) = (1)/(2) To find B, let x=-2: -2 = A(-2+2) + B(-2-2) -2 = 0 + B(-4) -2 = -4B B = (-2)/(-4) = (1)/(2) Substitute the values of A and B back into the decomposition: f(x) = (1)/(2)x-2 + (1)/(2)x+2 f(x) = (1)/(2(x-2)) + (1)/(2(x+2)) The partial fraction decomposition is (1)/(2(x-2)) + (1)/(2(x+2)). Send me the next one 📸