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^3 - 3x^2 + 4x -4; k=2

Verified step by step guidance

1. Write down the coefficients of the polynomial $f(x)=x^3-3x^2+4x-4$, which are [1, -3, 4, -4].
2. Set up the synthetic division by writing the value of $k=2$ to the left and the coefficients 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 under the next coefficient (-3).
5. Add the numbers in the second column (-3 and 2) and write the result in the bottom row. Repeat the multiply and add process for the remaining columns.

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 the value of k to perform the division in a more efficient manner than traditional long division. This technique helps in quickly determining the remainder, which indicates whether k is a zero of the polynomial.

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 is ƒ(x) = x^3 - 3x^2 + 4x - 4, which is a cubic polynomial. Understanding the structure of polynomial functions is essential for analyzing their roots and behavior.

Zero of a Polynomial

A zero of a polynomial is a value of x for which the polynomial evaluates to zero, meaning ƒ(k) = 0. Finding zeros is crucial for understanding the roots of the polynomial, which can indicate where the graph intersects the x-axis. If k is not a zero, calculating ƒ(k) provides the actual value of the polynomial at that point.