Problem 45 - A Graphical Approach to College Algebra

Question

Use synthetic division to determine whether the given number is a zero of the polynomial. $$2 ; \quad P(x)=x^{2}+2 x-8$$

Step-by-Step Solution

Answer

The number 2 is a zero of the polynomial, since the remainder is 0.

Step 1: Set Up the Division

To use synthetic division, we set up the coefficients of the polynomial based on \( P(x) = x^2 + 2x - 8 \). Write the coefficients in order: 1, 2, and -8. Place the number 2 (the possible zero) outside.

Step 2: Perform Synthetic Division

Start the division process. First, bring down the leading coefficient 1. Multiply 1 by 2 and write the result under the next coefficient. Add 2 and 2 to get 4. Repeat the process: multiply 4 by 2 resulting in 8, and write it under -8. Then, add: -8 + 8 = 0.

Step 3: Interpret the Remainder

After performing the synthetic division, observe the remainder, which is the last value after the division process. If the remainder is 0, then 2 is a zero of the polynomial. In this case, the remainder is 0.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression that involves a sum of powers of one or more variables multiplied by coefficients. These functions can be written as:\[P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0\]where:

• - \(a_n, a_{n-1}, \ldots, a_0\) are coefficients
• - \(x\) is the variable
• - \(n\) is a non-negative integer representing the degree of the polynomial

In the given exercise, the polynomial \(P(x) = x^2 + 2x - 8\) is a quadratic polynomial, which means its degree is 2 (the highest power of \(x\) is 2). Understanding the structure of polynomial functions is crucial because it helps us to analyze their properties, such as the number of roots or zeros they might have.

Zeros of Polynomials

The zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. In other words, if \(P(x) = 0\), then \(x\) is a zero of the polynomial. Finding these zeros helps in solving polynomial equations and understanding the roots where the graph of the polynomial touches or crosses the x-axis.

To determine if a number is a zero of a polynomial, we can use various methods. One efficient technique is synthetic division, as shown in the exercise. This technique helps us quickly check if a given number, like 2 in the exercise, is a zero by performing calculations directly with the coefficients of the polynomial.

In our example, after using synthetic division, we find the remainder to be 0, meaning that 2 is, indeed, a zero of the polynomial \(P(x) = x^2 + 2x - 8\). This tells us that when \(x=2\), the polynomial evaluates to zero (i.e., \(P(2) = 0\)).

Remainder Theorem

The Remainder Theorem is a key concept in algebra that relates the remainder of a polynomial division to its function value. Specifically, when a polynomial \(P(x)\) is divided by \(x - c\), the remainder of this division is equal to \(P(c)\).

• This means that if we substitute \(c\) into the polynomial and calculate, the result is the same as the remainder after performing the division.

This theorem is highly useful for evaluating polynomials at specific points or for determining zeros. If the remainder is 0, as shown in our synthetic division exercise, this confirms that the tested number is a zero of the polynomial.

By applying the Remainder Theorem during synthetic division, you can quickly verify potential zeros without performing full polynomial expansions or manually checking each substitution.