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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Step 1: Apply the Factor Theorem for f(x) with factor (x-1). Given f(x) = x^3 - hx^2 - 13x + k. If (x-1) is a factor, then f(1) = 0. f(1) = (1)^3 - h(1)^2 - 13(1) + k = 0 1 - h - 13 + k = 0 -h + k - 12 = 0 k - h = 12 (*) Step 2: Apply the Factor Theorem for f(x) with factor (x+3). If (x+3) is a factor, then f(-3) = 0. f(-3) = (-3)^3 - h(-3)^2 - 13(-3) + k = 0 -27 - 9h + 39 + k = 0 -9h + k + 12 = 0 k - 9h = -12 (**) Step 3: Solve the system of linear equations for h and k. Subtract equation (*) from equation (): (k - h) - (k - 9h) = 12 - (-12) k - h - k + 9h = 12 + 12 8h = 24 h = (24)/(8) h = 3 Step 4: Substitute h=3 into equation (*) to find k. k - 3 = 12 k = 12 + 3 k = 15 The values are h=3 and k=15. h=3, k=15 23. Step 1: Analyze p(x) with factor (x+4). Given p(x) = (x^2 - 3x + 2)(x^2 + 7x + a). If (x+4) is a factor of p(x), then p(-4) = 0. p(-4) = ((-4)^2 - 3(-4) + 2)((-4)^2 + 7(-4) + a) = 0 (16 + 12 + 2)(16 - 28 + a) = 0 (30)(-12 + a) = 0 Since 30 ≠ 0, we must have: -12 + a = 0 a = 12 Step 2: Analyze q(x) with factor (x-1). Given q(x) = (x^2 + 5x + 4)(x^2 - 5x + b). If (x-1) is a factor of q(x), then q(1) = 0. q(1) = ((1)^2 + 5(1) + 4)((1)^2 - 5(1) + b) = 0 (1 + 5 + 4)(1 - 5 + b) = 0 (10)(-4 + b) = 0 Since 10 ≠ 0, we must have: -4 + b = 0 b = 4 The values are a=12 and b=4. a=12, b=4 Send me the next one 📸