Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = x² +2x -8; k=2

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = x^2 + 2x - 8. These are 1 (for x^2), 2 (for x), and -8 (constant term).
2. Set up the synthetic division by writing the number k = 2 to the left, and the coefficients 1, 2, and -8 in a row to the right.
3. Bring down the first coefficient (1) directly below the line. Then multiply this number by k (2) and write the result under the next coefficient.
4. Add the numbers in the second column (2 + the product from the previous step), write the sum below the line, then repeat the multiply and add process for the last coefficient.
5. The final number you get after the last addition is the remainder, which equals ƒ(k). If this remainder is 0, then k is a zero of the polynomial; if not, this remainder is the value of ƒ(k).

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form (x - k). It simplifies the long division process by using only the coefficients of the polynomial, making it faster and less error-prone. This method helps determine the remainder when the polynomial is evaluated at k.

Zeros of a Polynomial

A zero of a polynomial is a value of x that makes the polynomial equal to zero. If k is a zero, then (x - k) is a factor of the polynomial, and the remainder after division by (x - k) is zero. Identifying zeros is essential for factoring and solving polynomial equations.

Evaluating Polynomial Functions

Evaluating a polynomial function at a specific value k means substituting k into the polynomial and calculating the result. If the result is zero, k is a root; otherwise, the value gives the output of the function at k. Synthetic division provides a quick way to find this value.