Problem 46 - Precalculus : Building Concepts and Connections

Question

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree 5 ; zeros at -3 and 1 , each of multiplicity \(2 ;\) zero at 4 of multiplicity 1

Step-by-Step Solution

Answer

The expression for the polynomial function given is \(f(x) = x^5 - 9x^4 + 47x^3 - 115x^2 + 144x - 72\).

Step 1: Consider root -3 with multiplicity 2

The factor corresponding to root -3 with multiplicity 2 will be \((x+3)^2\). Multiplicity refers to how many times a particular number is a zero for a given polynomial.

Step 2: Consider root 1 with multiplicity 2

The factor corresponding to root 1 with multiplicity 2 will be \((x-1)^2\).

Step 3: Consider root 4 with multiplicity 1

The factor corresponding to root 4 with multiplicity 1 will be \((x-4)\).

Step 4: Construct the polynomial function

These factors combined will give the polynomial function \(f(x) = (x+3)^2 * (x-1)^2 * (x-4)\). This polynomial has degree 5, as required.

Step 5: Simplify the polynomial function

Multiplying and simplifying these factors give the simplified polynomial function \(f(x) = x^5 - 9x^4 + 47x^3 - 115x^2 + 144x - 72\).

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is fundamental when studying polynomial functions. The degree of a polynomial is the highest power of the variable in the expression. It indicates the maximum number of possible roots or solutions a polynomial equation can have, also informing us about the graph's end behavior.

For example, in the given exercise, the polynomial is of degree 5, as indicated by the final polynomial expression's highest power, which is 5. This degree identifies it as a quintic polynomial. It suggests that the polynomial function will have a maximum of five roots or solutions.

Additionally, the degree determines the leading term's impact on the graph, especially for large values of the variable. In our polynomial, the leading term is \(x^5\), meaning the ends of the graph will behave like the function \(y = x^5\), which starts high and ends high if the coefficient is positive.

Multiplicity of Roots

The multiplicity of roots plays a crucial role in defining the structure of a polynomial function. Multiplicity refers to the number of times a root is repeated in a polynomial. It affects the graph's appearance and how it touches or crosses the x-axis.

In the given problem, the roots have different multiplicities, indicating how they will interact with the axis:

• The root \(-3\) has a multiplicity of 2. This means the graph will touch the x-axis at \(-3\) and "bounce off" without crossing it.
• The root \(1\) also has a multiplicity of 2, resulting in similar behavior at this point as well.
• The root \(4\), having a multiplicity of 1, indicates that the polynomial will cross the x-axis at this point.

Understanding these interactions helps in sketching and analyzing the graph of the polynomial. A higher multiplicity at a root point creates a smoother turn at that location on the graph.

Zero of a Polynomial

Zeros of a polynomial, often called roots or solutions, are essential in the study of polynomial functions. These are the values of \(x\) that make the polynomial equal to zero. Identifying zeros helps in forming the factors of the polynomial and in sketching the graph.

In the exercise, the zeros are explicitly given with their multiplicities, which helps us directly form the polynomial using zero-factor pairs or factors:

• A root of \(-3\) with multiplicity 2 leads to a factor of \((x+3)^2\).
• The root \(1\) with multiplicity 2 results in the factor \((x-1)^2\).
• A root of \(4\) with multiplicity 1 gives us the factor \((x-4)\).

By multiplying these factors, we build the polynomial function, each zero contributing to the overall structure of the equation. Notably, zeros with higher multiplicities indicate repeating factors and specific behaviors at those x-values on the graph.