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 +7x^2 + 10x; k=0

Verified step by step guidance

1. Write down the coefficients of the polynomial \( f(x) = x^3 + 7x^2 + 10x \). These are [1, 7, 10, 0].
2. Set up the synthetic division by writing the value of \( k = 0 \) 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 by \( k \) (which is 0) and write the result under the next coefficient.
5. Add the numbers in the second column 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 allows for quick calculations of polynomial values and helps determine if k is a root of the polynomial. This method involves using the coefficients of the polynomial and performing a series of arithmetic operations, making it more efficient than traditional long division.

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 + 7x^2 + 10x, which is a cubic polynomial. Understanding the degree and behavior of polynomial functions is essential for analyzing their roots and values at specific points.

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, which is important for further analysis.