Given the function \(f(x) = x^4 - 3x^3 + 2x^2 - x - 1\). Find the derivative of the function using synthetic division and algebraic concepts.
Answer
Step 5: Multiply the derivative by the factor (x-1) to get the derivative of the original function: \((x - 1)(3x^2 - 4x - 1) = 3x^3 - 7x^2 + 4x - 1\).
Step 1 ：Step 1: We start by finding the roots of the given polynomial by using the synthetic division. We try the factors of the constant term -1, which are ±1. Let's try 1 first.
Step 2 ：Step 2: Set up the synthetic division tableau: \[\begin{array}{c|ccccc} 1 & 1 & -3 & 2 & -1 & -1 \end{array}\]
Step 3 ：Step 3: Perform the synthetic division: \[\begin{array}{c|ccccc} 1 & 1 & -2 & 0 & -1 & 0 \end{array}\]. So, x-1 is a factor of the polynomial, and the remaining polynomial is \(x^3 - 2x^2 - x\).
Step 4 ：Step 4: Differentiate the remaining polynomial: \(\frac{d}{dx}(x^3 - 2x^2 - x) = 3x^2 - 4x - 1\).
Step 5 ：Step 5: Multiply the derivative by the factor (x-1) to get the derivative of the original function: \((x - 1)(3x^2 - 4x - 1) = 3x^3 - 7x^2 + 4x - 1\).
