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.
ScanSolve AI Answer
Welcome back ogbodofavour890 — missed you this week. Here's the solution to question 3: a) To show that (x+2) is a factor of f(x), we use the Factor Theorem. If (x+2) is a factor, then f(-2) must be equal to 0. Step 1: Substitute x = -2 into f(x). f(x) = x^3 - 2x^2 - 5x + 6 f(-2) = (-2)^3 - 2(-2)^2 - 5(-2) + 6 Step 2: Calculate the value of f(-2). f(-2) = -8 - 2(4) + 10 + 6 f(-2) = -8 - 8 + 10 + 6 f(-2) = -16 + 16 f(-2) = 0 Since f(-2) = 0, (x+2) is indeed a factor of f(x). b) To factorise f(x) completely, we can use polynomial division or synthetic division since we know (x+2) is a factor. Step 1: Perform polynomial division of f(x) by (x+2). (x^3 - 2x^2 - 5x + 6) ÷ (x+2) Using synthetic division with root -2: ` -2 | 1 -2 -5 6 | -2 8 -6 ------------------ 1 -4 3 0 ` The quotient is x^2 - 4x + 3. Step 2: Factorise the quadratic quotient. x^2 - 4x + 3 We need two numbers that multiply to 3 and add to -4. These numbers are -1 and -3. (x-1)(x-3) Step 3: Write the complete factorisation of f(x). f(x) = (x+2)(x-1)(x-3) The complete factorisation of f(x) is (x+2)(x-1)(x-3). Drop the next question 📸