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) = 4x^4 + x^2 + 17x + 3; k= -3/2

Verified step by step guidance

1. Write down the coefficients of the polynomial \( f(x) = 4x^4 + 0x^3 + 1x^2 + 17x + 3 \) as \([4, 0, 1, 17, 3]\).
2. Set up the synthetic division by writing \( k = -\frac{3}{2} \) to the left and the coefficients to the right.
3. Bring down the leading coefficient (4) to the bottom row.
4. Multiply \(-\frac{3}{2}\) by the number just written on the bottom row (4) and write the result under the next coefficient (0).
5. Add the numbers in the second column and write the result in the bottom row. Repeat the multiply and add process for each column until you reach the last coefficient.

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 performing a series of multiplications and additions to find the quotient and remainder. This technique is particularly useful for evaluating polynomials at specific values and determining if those values are roots 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. The general form of a polynomial in one variable x is f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where a_n are constants and n is a non-negative integer. Understanding the structure of polynomial functions is essential for analyzing their behavior, including finding zeros and evaluating function values.

Zero of a Polynomial

A zero of a polynomial is a value of x for which the polynomial evaluates to zero, meaning f(k) = 0. Finding zeros is crucial for understanding the roots of the polynomial, which can indicate where the graph intersects the x-axis. If a given number k is not a zero, evaluating the polynomial at k provides the corresponding function value, which helps in analyzing the polynomial's behavior at that point.