Use synthetic division to find ƒ(2). ƒ(x)=x^5+4x^2-2x-4

Verified step by step guidance

1. Identify the polynomial function \( f(x) = x^5 + 4x^2 - 2x - 4 \) and the value \( x = 2 \) for which you want to find \( f(2) \) using synthetic division.
2. Set up the synthetic division by writing down the coefficients of the polynomial: \( 1, 0, 0, 4, -2, -4 \). Note that you include zeros for any missing terms (\( x^4 \) and \( x^3 \) in this case).
3. Write the value \( 2 \) to the left of the coefficients, which is the value you are evaluating the function at.
4. Bring down the leading coefficient (1) to the bottom row. Multiply this number by 2 (the value you are evaluating at) and write the result under the next coefficient. Add this result to the next coefficient and write the sum below the line.
5. Continue this process of multiplying the result by 2 and adding to the next coefficient until you reach the end. The final number at the bottom right is \( f(2) \).

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial long division that allows for the efficient division of a polynomial by a linear factor of the form (x - c). It involves using the coefficients of the polynomial and performing a series of multiplications and additions to find the quotient and remainder. This method is particularly useful for evaluating polynomials at specific values, such as finding ƒ(2) in this case.

Polynomial Evaluation

Polynomial evaluation is the process of calculating the value of a polynomial function at a specific input. For a polynomial ƒ(x), substituting a value for x allows us to determine the output of the function. In this question, we are tasked with evaluating the polynomial ƒ(x) = x^5 + 4x^2 - 2x - 4 at x = 2, which can be efficiently done using synthetic division.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is equal to f(c). This theorem provides a quick way to evaluate polynomials at specific points without fully performing the division. In this context, using synthetic division to find ƒ(2) will yield the remainder, which directly gives us the value of the polynomial at that point.