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) = 2x³ - 6x² -9x + 4; k=1

Verified step by step guidance

1. Write down the coefficients of the polynomial ƒ(x) = 2x^3 - 6x^2 - 9x + 4. These are 2, -6, -9, and 4.
2. Set up synthetic division by writing the number k = 1 to the left, and the coefficients in a row to the right: 2, -6, -9, 4.
3. Bring down the first coefficient (2) as it is. Then multiply this number by k (1) and write the result under the next coefficient.
4. Add the column: add the second coefficient (-6) and the number just written. Write the sum below the line. Repeat the multiply and add process for all coefficients.
5. The last number you get after adding is the remainder, which equals ƒ(k). If this remainder is 0, then k is a zero of the polynomial; if not, the remainder is the value of ƒ(k).

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - k). It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone. This method helps determine the remainder when the polynomial is divided by (x - 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 when dividing 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 zero of the polynomial. If not, the value obtained is the remainder when dividing by (x - k), which synthetic division can also provide.