Use synthetic division to find ƒ(2). ƒ(x)=x⁵+4x²-2x-4

Verified step by step guidance

1. Identify the polynomial function: \(f(x) = x^5 + 4x^2 - 2x - 4\). Note that some terms are missing (like \(x^4\), \(x^3\), and the constant term), so include them with zero coefficients for synthetic division: \(x^5 + 0x^4 + 0x^3 + 4x^2 - 2x - 4\).
2. Set up synthetic division using the value \(x = 2\). Write down the coefficients in order: \([1, 0, 0, 4, -2, -4]\) and place the number 2 to the left.
3. Bring down the first coefficient (1) as is. Multiply it by 2 and write the result under the next coefficient. Add the column and write the sum below. Repeat this multiply-and-add process for all coefficients.
4. The last number you obtain after completing the synthetic division process is the remainder, which equals \(f(2)\) according to the Remainder Theorem.
5. Interpret the remainder as the value of the function at \(x=2\), which is \(f(2)\). This completes the process of finding \(f(2)\) using synthetic division.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone.

Evaluating Polynomials Using Synthetic Division

Synthetic division can be used to evaluate a polynomial at a specific value by dividing the polynomial by (x - c) and examining the remainder. The remainder from this division equals the value of the polynomial at x = c.

Polynomial Coefficients and Missing Terms

When performing synthetic division, all powers of x must be accounted for, including those with zero coefficients. For example, if a term like x^4 or x^3 is missing, its coefficient is zero and must be included in the synthetic division setup.