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: Find the value of k using the Factor Theorem. If (x-1) is a factor of f(x), then f(1) = 0. Substitute x=1 into the polynomial f(x) = 4x^3 - 4x^2 - x - k: f(1) = 4(1)^3 - 4(1)^2 - (1) - k 0 = 4(1) - 4(1) - 1 - k 0 = 4 - 4 - 1 - k 0 = -1 - k k = -1 Step 2: Substitute the value of k back into the polynomial. The polynomial becomes: f(x) = 4x^3 - 4x^2 - x - (-1) f(x) = 4x^3 - 4x^2 - x + 1 Step 3: Factorize the polynomial using synthetic division. Since (x-1) is a factor, we divide f(x) by (x-1). Using synthetic division with the root x=1: ` 1 | 4 -4 -1 1 | 4 0 -1 ---------------- 4 0 -1 0 ` The quotient is 4x^2 + 0x - 1, which simplifies to 4x^2 - 1. So, f(x) = (x-1)(4x^2 - 1). Step 4: Factorize the quadratic expression completely. The quadratic expression 4x^2 - 1 is a difference of squares, a^2 - b^2 = (a-b)(a+b), where a=2x and b=1. 4x^2 - 1 = (2x)^2 - (1)^2 = (2x - 1)(2x + 1) Step 5: Write the completely factorized polynomial. Substitute the factored quadratic back into the expression for f(x): f(x) = (x-1)(2x-1)(2x+1) The completely factorized polynomial is f(x) = (x-1)(2x-1)(2x+1). Step 6: State the zeros of the polynomial. To find the zeros, set each factor equal to zero and solve for x: For the first factor: x-1 = 0 x = 1 For the second factor: 2x-1 = 0 2x = 1 x = (1)/(2) For the third factor: 2x+1 = 0 2x = -1 x = -(1)/(2) The zeros of the polynomial are 1, (1)/(2), -(1)/(2). That's 2 down. 3 left today — send the next one.