Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polynomial function with the given zeros, multiplicities, and degree. (There are many correct answers.) Zero: \(-1,\) multiplicity: 2 Zero: \(-2,\) multiplicity: 1 Degree: 3 Rises to the left, Falls to the right

Step-by-Step Solution

Answer

The polynomial function with the given zeros, multiplicities, and degree, that rises to the left and falls to the right is \(-x^3 - 4x^2 - 5x - 2\).

Step 1: Identify the zeros and their multiplicities

Given zeros are -1 and -2, with multiplicities of 2 and 1 respectively.

Step 2: Formulate the basic equation

Based on the provided zeros and multiplicities, we can determine that the function will have factors of the form \((x-(-1))^2\) and \((x-(-2))^1\), because the zero at x=-1 has multiplicity 2 and the zero at x=-2 has multiplicity 1.

Step 3: Multiply out the basic equation

This result in the equation \((x+1)^2 * (x+2)\), which multiplies out to \(x^3 + 4x^2 + 5x + 2\).

Step 4: Incorporate direction of graph

Because the direction of the polynomial 'rises to the left and falls to the right', the leading coefficient of the polynomial must be negative. Therefore, we multiply the entire function by -1, resulting in \(-x^3 - 4x^2 - 5x - 2\).

Key Concepts

Roots and Zeros

In a polynomial function, finding the roots or zeros is like finding the places where the function touches or crosses the x-axis. These are the values of the variable for which the polynomial equals zero. For example, in the given exercise, the zeros are \(-1\) and \(-2\). Simply put, these are the values where the polynomial function will have an output of zero.

When you plot a polynomial function on a graph, the zeros are the x-values where the curve meets or crosses the x-axis. Knowing the zeros helps us understand the behavior of the polynomial and provides a foundation for constructing the polynomial expression.

In practice:

• Zeros are solutions to the polynomial equation when set to zero.
• They help in factoring the polynomial into simpler expressions.
• Identifying zeros is a key step in sketching the graph of the polynomial.

Understanding zeros is crucial because they not only tell us about the graph's behavior but also lead us toward finding possible points of intersection with the x-axis.

Multiplicity

Multiplicity refers to how many times a particular root is repeated in a polynomial function. It's an important concept because it affects the graph's shape and the function's behavior at the zeros. In our exercise, we have a zero at \(-1\) with a multiplicity of 2, and a zero at \(-2\) with a multiplicity of 1.

Higher multiplicities mean that the graph will "bounce" off or "touch" the x-axis at that root, rather than fully crossing it. This can influence how the polynomial graph behaves near those points.

To interpret multiplicities:

• A root with an odd multiplicity implies that the graph crosses the x-axis at the zero.
• A root with an even multiplicity means the graph just touches or bounces off the x-axis.
• Multiplicity of a root is an exponent indicating how many times the root occurs in the factored form of the polynomial.

In this case, the root \(-1\) with multiplicity 2 means the graph will touch the x-axis at \(-1\) and bounce back. Meanwhile, the root \(-2\) with multiplicity 1 indicates the graph will cross the x-axis at \(-2\). Understanding multiplicity gives insight into the smoothness and turning points of the graph at each zero.

Graph Behavior

Graph behavior of a polynomial function is largely influenced by the degree and leading coefficient of the polynomial. In the exercise, the polynomial \[-x^3 - 4x^2 - 5x - 2\] is a cubic function (degree 3) with a negative leading coefficient. This set of properties gives us essential clues on how the graph will behave at the ends.

The degree of the polynomial tells us about the graph's general shape and how many extrema (high or low points) it may have. For cubic polynomials:

• They can have up to 2 turning points.
• The behavior at infinity is determined by the leading term \(-x^3\).
• A negative leading coefficient implies the graph rises on one side and falls on the other.

In our specific case:

• "Rises to the left, falls to the right" suggests that as we move to the left (towards negative infinity), the graph will rise higher up, and as we move to the right (towards positive infinity), the graph will fall lower.
• The zeros \(-1\) and \(-2\) with their respective multiplicities influence local behavior but not the end behavior.
• The graph will simply touch at the zero \(-1\) and cross the axis at \(-2\), then proceed to follow the general direction dictated by its leading term.

This behavior is crucial for sketching the graph accurately, providing a solid understanding of how the polynomial function looks visually.