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
Step 1: Apply the Factor Theorem. The Factor Theorem states that if (x-c) is a factor of a polynomial F(x), then F(c) = 0. Given that (x-3) is a factor of F(x) = ax^3 + bx^2 - 9x - 9, we must have F(3) = 0. Step 2: Substitute x=3 into F(x). Substitute x=3 into the polynomial F(x): F(3) = a(3)^3 + b(3)^2 - 9(3) - 9 Step 3: Set F(3) to zero and simplify the equation. Since F(3) = 0: a(27) + b(9) - 27 - 9 = 0 27a + 9b - 36 = 0 Divide the entire equation by 9: (27a)/(9) + (9b)/(9) - (36)/(9) = (0)/(9) 3a + b - 4 = 0 Rearrange the equation to match the required form: 3a + b = 4 This shows that 3a + b = 4. b) When F(x) is divided by (x-2), the remainder is -15. This is a given condition. By the Remainder Theorem, this means F(2) = -15. Send me the next one 📸