Problem 7 - Precalculus : Building Concepts and Connections

Question

For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$g(s)=s^{2}+4, s=-2,2$$

Step-by-Step Solution

Answer

Neither \(s = -2\) nor \(s = 2\) are zeros of the polynomial \(g(s) = s^2 + 4\).

Step 1: Substitute the first value

Substitute \(s = -2\) into \(g(s) = s^2 + 4\). So, we get \(g(-2) = (-2)^2 + 4 = 4 + 4 = 8\). Thus, \(g(-2) ≠ 0\), which implies \(s = -2\) is not a zero of the polynomial.

Step 2: Substitute the second value

Substitute \(s = 2\) into \(g(s) = s^2 + 4\). So, we get \(g(2) = (2)^2 + 4 = 4 + 4 = 8\). Again, \(g(2) ≠ 0\), which implies \(s = 2\) is not a zero of the polynomial.

Key Concepts

Polynomial Functions

When we talk about polynomial functions, we're looking at a type of mathematical expression that involves a sum of powers of a variable, typically denoted as 'x' or 's'. These powers are all non-negative integers, and the coefficients can be any real or complex number. For example, the quadratic function f(x) = ax^2 + bx + c is a polynomial, where a, b, and c are constants.

Polynomial functions can have various degrees, with the highest exponent of the variable indicating the degree. For instance, in g(s) = s^2 + 4, the highest power of s is 2, so it is called a second-degree polynomial or a quadratic polynomial. Whether it's a second-degree polynomial or a higher degree, understanding the structure of these functions helps us analyze their properties, such as their zeros, end behavior, and graph shape.

Substitution Method

The substitution method is a straightforward technique we use in algebra to evaluate functions or solve equations. It involves replacing a variable with a given number to compute the function's value or to simplify the expression. This is especially useful when looking for zeros of a polynomial or when trying to find specific values that satisfy an equation.

To apply the substitution method correctly, it's important to ensure that every instance of the variable is replaced consistently throughout the expression. After substitution, the result will either confirm or disprove our initial assumption. For example, substituting s = -2 into g(s) = s^2 + 4, transform the expression into a numerical equation that can be solved easily, helping us to understand whether -2 is a zero of the polynomial.

Zeros of a Polynomial

Zeros of a polynomial are values for the variable that make the polynomial equal to zero. Essentially, they are the solutions to the equation f(x) = 0. Finding these zeros is crucial because they give us key insights into the behavior of the polynomial's graph, including its x-intercepts where it crosses the x-axis.

In the given exercise g(s) = s^2 + 4, the task was to determine if s = -2 or s = 2 are zeros of the polynomial. By using the substitution method, we concluded that neither of these values are zeros since substituting them into the equation did not result in a zero. This means for the quadratic g(s), the graph does not touch or cross the horizontal axis at s = -2 or s = 2, conveying that it doesn't have real roots. Understanding zeros can also lead to insights on factoring polynomials and solving polynomial equations.