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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1.1 Given the function: f(x) = -x^2 - 2x^2 + 5x + 6 Step 1: Simplify the function. f(x) = -3x^2 + 5x + 6 Step 2: To find the roots, set f(x) = 0. -3x^2 + 5x + 6 = 0 Step 3: Multiply by -1 to make the leading coefficient positive. 3x^2 - 5x - 6 = 0 Step 4: Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a with a=3, b=-5, c=-6. x = -(-5) ± sqrt((-5)^2 - 4(3)(-6))2(3) x = 5 ± sqrt(25 + 72)6 x = 5 ± sqrt(97)6 The final answers are: x = 5 + sqrt(97)6 or x = 5 - sqrt(97)6 1.2 Given the function f(x) = x^3 + ax^2 + bx + 8 is divisible by x^2 - x - 2. Step 1: Factorize the divisor x^2 - x - 2. x^2 - x - 2 = (x-2)(x+1) Step 2: Since f(x) is divisible by (x-2)(x+1), it means x=2 and x=-1 are roots of f(x). Therefore, f(2) = 0 and f(-1) = 0. Step 3: Substitute x=2 into f(x) and set it to 0. f(2) = (2)^3 + a(2)^2 + b(2) + 8 = 0 8 + 4a + 2b + 8 = 0 4a + 2b + 16 = 0 Divide by 2: 2a + b + 8 = 0 2a + b = -8 (Equation 1) Step 4: Substitute x=-1 into f(x) and set it to 0. f(-1) = (-1)^3 + a(-1)^2 + b(-1) + 8 = 0 -1 + a - b + 8 = 0 a - b + 7 = 0 a - b = -7 (Equation 2) Step 5: Solve the system of linear equations (Equation 1 and Equation 2). Add Equation 1 and Equation 2: (2a + b) + (a - b) = -8 + (-7) 3a = -15 a = -5 Step 6: Substitute a=-5 into Equation 2 to find b. -5 - b = -7 -b = -7 + 5 -b = -2 b = 2 The final answers are: a = -5, b = 2 1.3 Given the function f(x) = x^3 - kx^2 - x + 13. When divided by x-3, the remainder is 10. Step 1: Apply the Remainder Theorem, which states that if a polynomial f(x) is divided by x-c, the remainder is f(c). In this case, c=3 and the remainder is 10, so f(3) = 10. Step 2: Substitute x=3 into f(x) and set it equal to 10. f(3) = (3)^3 - k(3)^2 - (3) + 13 = 10 27 - 9k - 3 + 13 = 10 Step 3: Simplify and solve for k. 37 - 9k = 10 -9k = 10 - 37 -9k = -27 k = (-27)/(-9) k = 3 The final answer is: k = 3 3 done, 2 left today. You're making progress.