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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Here are the solutions to the questions. Question 2: The polynomial f(x) = ax^3 - x^2 + bx + 6. (a) Find the values of the constants a and b. Step 1: Use the Factor Theorem. Since (x-2) is a factor of f(x), we know that f(2) = 0. Substitute x=2 into the polynomial: f(2) = a(2)^3 - (2)^2 + b(2) + 6 = 0 8a - 4 + 2b + 6 = 0 8a + 2b + 2 = 0 Divide the equation by 2: 4a + b + 1 = 0 4a + b = -1 (Equation 1) Step 2: Use the Remainder Theorem. When f(x) is divided by (x+1), the remainder is 10. This means f(-1) = 10. Substitute x=-1 into the polynomial: f(-1) = a(-1)^3 - (-1)^2 + b(-1) + 6 = 10 -a - 1 - b + 6 = 10 -a - b + 5 = 10 -a - b = 5 a + b = -5 (Equation 2) Step 3: Solve the system of linear equations for a and b. We have: 1) 4a + b = -1 2) a + b = -5 Subtract Equation 2 from Equation 1: (4a + b) - (a + b) = -1 - (-5) 3a = 4 a = (4)/(3) Substitute the value of a into Equation 2: (4)/(3) + b = -5 b = -5 - (4)/(3) b = -(15)/(3) - (4)/(3) b = -(19)/(3) The values of the constants are a = (4)/(3), b = -(19)/(3). (b) Find the values of x for which f(x) = 0. Step 1: Write the polynomial with the found values of a and b. f(x) = (4)/(3)x^3 - x^2 - (19)/(3)x + 6 To make calculations easier, we can multiply the entire equation by 3 (since we are looking for roots, f(x)=0 implies 3f(x)=0): 3f(x) = 4x^3 - 3x^2 - 19x + 18 Step 2: Use synthetic division to find the quadratic factor. Since (x-2) is a factor, x=2 is a root. We can use synthetic division with 2: ` 2 | 4 -3 -19 18 | 8 10 -18 -------------------- 4 5 -9 0 ` The quotient is 4x^2 + 5x - 9. So, f(x) = (x-2)(4x^2 + 5x - 9) = 0. Step 3: Solve the quadratic equation 4x^2 + 5x - 9 = 0. We can factor this quadratic equation. We look for two numbers that multiply to 4 × (-9) = -36 and add to 5. These numbers are 9 and -4. 4x^2 + 9x - 4x - 9 = 0 x(4x+9) - 1(4x+9) = 0 (x-1)(4x+9) = 0 Step 4: Find the roots from the factors. From (x-2)=0, we get x=2. From (x-1)=0, we get x=1. From (4x+9)=0, we get 4x = -9 x = -(9)/(4). The values of x for which f(x)=0 are x = 2, x = 1, x = -(9)/(4). --- Question 3: The question is incomplete as the full table of values for y is not provided, and the specific task (e.g., "plot a graph", "find an equation", "estimate a value") is missing. Please provide the complete question and table for a solution.