Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = x^2 +2x -8; k=2

Verified step by step guidance

1. Write down the coefficients of the polynomial \( f(x) = x^2 + 2x - 8 \), which are [1, 2, -8].
2. Set up the synthetic division by writing the value of \( k = 2 \) to the left and the coefficients [1, 2, -8] to the right.
3. Bring down the leading coefficient (1) to the bottom row.
4. Multiply the value just written on the bottom row (1) by \( k = 2 \) and write the result (2) under the next coefficient (2).
5. Add the numbers in the second column (2 + 2) and write the result (4) in the bottom row. Repeat the multiplication and addition process for the next column.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear binomial of the form (x - k). It involves using the coefficients of the polynomial and performing a series of multiplications and additions to find the quotient and remainder. This technique is particularly useful for quickly determining if a given value k is a root of the polynomial, as the remainder will indicate whether ƒ(k) equals zero.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, the polynomial function is ƒ(x) = x² + 2x - 8, which is a quadratic function. Understanding the structure of polynomial functions is essential for analyzing their roots, behavior, and how they can be evaluated at specific points.

Evaluating a Function

Evaluating a function involves substituting a specific value for the variable in the function's expression to find the corresponding output. For the polynomial function ƒ(x), evaluating it at k = 2 means calculating ƒ(2) = 2² + 2(2) - 8. This process is crucial for determining whether k is a zero of the polynomial, as a zero indicates that the function's value at that point is zero.