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
To solve problem 12, we will use the Factor Theorem. 12. Find b if h(x) = 2x-1 is a factor of f(x) = 6x^2 + bx + 2. Step 1: Find the root of the factor h(x). If h(x) = 2x-1 is a factor, then h(x)=0 for some value of x. 2x - 1 = 0 2x = 1 x = (1)/(2) Step 2: Apply the Factor Theorem. According to the Factor Theorem, if (x-k) is a factor of a polynomial f(x), then f(k)=0. In this case, k = (1)/(2), so f((1)/(2)) must be equal to 0. Substitute x = (1)/(2) into f(x) = 6x^2 + bx + 2: f((1)/(2)) = 6((1)/(2))^2 + b((1)/(2)) + 2 = 0 Step 3: Solve the equation for b. 6((1)/(4)) + (b)/(2) + 2 = 0 (6)/(4) + (b)/(2) + 2 = 0 (3)/(2) + (b)/(2) + 2 = 0 Multiply the entire equation by 2 to eliminate the denominators: 2((3)/(2)) + 2((b)/(2)) + 2(2) = 2(0) 3 + b + 4 = 0 b + 7 = 0 b = -7 The value of b is -7.