Use synthetic division to find ƒ(2). ƒ(x)=5x⁴-12x²+2x-8

Verified step by step guidance

1. Identify the polynomial function: \(f(x) = 5x^4 - 12x^2 + 2x - 8\). Note that the polynomial is missing the \(x^3\) term, so include it with a coefficient of 0 for synthetic division.
2. Set up the synthetic division by writing the coefficients of \(f(x)\) in descending order of powers, including the zero for the missing term: \([5, 0, -12, 2, -8]\).
3. Write the value of \(x\) at which you want to evaluate the function, which is 2, to the left of the synthetic division setup.
4. Perform synthetic division: bring down the first coefficient, multiply it by 2, add to the next coefficient, and repeat this process across all coefficients.
5. The final number obtained after completing synthetic division is the value of \(f(2)\).

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 Terms

Understanding the coefficients and terms of a polynomial is essential for synthetic division. Each term's coefficient is used in the synthetic division process, and missing terms must be represented with zero coefficients to maintain the correct order.