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) = x³ - 3x² + 4x -4; k=2

Verified step by step guidance

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

Key Concepts

Polynomial Functions

A polynomial function is an expression involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding the structure of polynomials, such as degree and coefficients, is essential for evaluating and manipulating them.

Synthetic Division

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

Zeros of a Polynomial

A zero of a polynomial is a value of x that makes the polynomial equal to zero. Determining if k is a zero involves evaluating the polynomial at k; if the result is zero, k is a root, otherwise, the value ƒ(k) indicates the polynomial's output at that point.