Problem 9 - Precalculus : Building Concepts and Connections

Question

For each polynomial function, list the zeros of the polynomial and state the multiplicity of each zero. $$f(s)=(s-\pi)^{10}(s+\pi)^{3}$$

Step-by-Step Solution

Answer

The zeros of the polynomial \( f(s)=(s-\pi)^{10}(s+\pi)^{3} \) are \(s = \pi\) and \(s = -\pi\). The zero \(s = \pi\) has multiplicity 10 and the zero \(s = -\pi\) has multiplicity 3.

Step 1: Identify the Zeros

To find the zeros of the polynomial, we need to set the polynomial equivalent to zero, and then solve for 's'. From the given polynomial \( f(s)=(s-\pi)^{10}(s+\pi)^{3} \), we can see that the solutions are \(s = \pi\) and \(s = -\pi\). This is because if we substitute \(s = \pi\) and \(s = -\pi\) into the equation, the polynomial is equal to zero.

Step 2: Identify the Multiplicity of each Zero

The multiplicity of a root or zero is the number of times it appears in the factored form of the polynomial. Looking at the form of the polynomial we can see that \( (s-\pi)^{10} \) and \( (s+\pi)^{3} \) are the factors. The exponent attached to each bracket represents the multiplicity of that root. Thus, the multiplicity of \(\pi\) is 10 and the multiplicity of -\(\pi\) is 3.

Key Concepts

Zeros of a Polynomial Function

The zeros of a polynomial function are the values for which the function equals zero. In simpler terms, they are the solutions to the equation formed by setting the polynomial equal to zero. For the polynomial \( f(s) = (s - \pi)^{10}(s + \pi)^{3} \), the zeros can be found by examining each part of the factorized equation separately. Each zero corresponds to a factor, specifically the point where a factor equals zero. For instance:

• The factor \((s - \pi)\) equals zero when \(s = \pi\).
• Likewise, the factor \((s + \pi)\) equals zero when \(s = -\pi\).

Thus, the zeros of this polynomial function are \(s = \pi\) and \(s = -\pi\). Finding zeros is a critical step as they often provide valuable insight into the behavior and graph of the polynomial.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears in the factored form of a polynomial. Multiplicity is associated with the exponent of the factor that is zeroed out. Let's break it down using the polynomial \( f(s) = (s - \pi)^{10}(s + \pi)^{3} \). Here’s how the multiplicity is determined:

• The zero \(s = \pi\) comes from the factor \((s - \pi)^{10}\). The exponent \(10\) indicates that \(\pi\) is a zero with multiplicity 10.
• The zero \(s = -\pi\) originates from the factor \((s + \pi)^{3}\). The exponent \(3\) indicates that \(-\pi\) is a zero with multiplicity 3.

Understanding multiplicity is important because it affects the polynomial's graph. For example, a zero with odd multiplicity will cross the x-axis, while a zero with even multiplicity will touch the x-axis and turn around.

Factored Form of a Polynomial

The factored form of a polynomial is an expression written as a product of its factors, making it simpler to identify zeros and their multiplicity. This form is fundamental when analyzing polynomials, as it reveals essential characteristics:

• For the polynomial \(f(s) = (s - \pi)^{10}(s + \pi)^{3}\), it is already in factored form. The components \((s - \pi)^{10}\) and \((s + \pi)^{3}\) each represent factors of the polynomial.
• Each factor equating to zero gives us a zero of the polynomial, while the exponents indicate the multiplicity of those zeros.

Using the factored form, you can quickly identify the zeros and understand their impact, such as how they influence the shape and intersection points on the graph. Factored form simplifies solving polynomial equations and helps provide clear insights into the nature of the function.