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: Use the Factor Theorem to confirm (x-1) is a factor. According to the Factor Theorem, if (x-1) is a factor of f(x), then f(1) must be equal to 0. Substitute x=1 into the polynomial f(x) = x^3 + 3x^2 - x - 3: f(1) = (1)^3 + 3(1)^2 - (1) - 3 f(1) = 1 + 3 - 1 - 3 f(1) = 4 - 4 f(1) = 0 Since f(1)=0, (x-1) is indeed a factor of f(x). Step 2: Perform polynomial division to find the other factor. We can use synthetic division to divide x^3 + 3x^2 - x - 3 by (x-1). The root is x=1. ` 1 | 1 3 -1 -3 | 1 4 3 ----------------- 1 4 3 0 ` The coefficients of the quotient are 1, 4, 3, and the remainder is 0. This means that x^3 + 3x^2 - x - 3 = (x-1)(x^2 + 4x + 3). Step 3: Factor the resulting quadratic expression. The quadratic expression is x^2 + 4x + 3. We need to find two numbers that multiply to 3 and add to 4. These numbers are 1 and 3. So, the quadratic factors as (x+1)(x+3). Step 4: Find all the zeros of f(x). Now we have the fully factored form of f(x): f(x) = (x-1)(x+1)(x+3) To find the zeros, set f(x)=0: (x-1)(x+1)(x+3) = 0 This gives us three possible solutions: x-1 = 0 x = 1 x+1 = 0 x = -1 x+3 = 0 x = -3 The zeros of f(x) are 1, -1, and -3. x = 1, x = -1, x = -3 That's 2 down. 3 left today — send the next one.