Problem 75 - 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: \(-2,\) multiplicity: 2 Zero: \(-1,\) multiplicity: 1 Degree: 3

Step-by-Step Solution

Answer

The polynomial function with the given zeros, multiplicities, and degree is \(P(x) = x^{3} + 5x^{2} + 8x + 4\).

Step 1: Understanding the properties of a polynomial

First, when zeros and their multiplicities are given for a polynomial, each pair of zero and multiplicity can contribute a factor of the form \( (x - zero)^{multiplicity} \) to the function. This is due to the property of polynomials that - a polynomial of degree ‘n’ has ‘n’ roots (or zeros).

Step 2: Apply the zeros and multiplicities to the polynomial

Considering all zeros and their multiplicities, construct the polynomial: The zero \(-2\) has multiplicity \(2\), hence \((x - (-2))^{2}\), which is \((x+2)^{2}\), The zero \(-1\) has multiplicity \(1\), hence \((x - (-1))^{1}\), which is \((x+1)\).

Step 3: Construct the polynomial

Multiplying these factors, the polynomial is given by \(P(x) = (x+2)^{2}*(x+1)\). Multiplying this out results in \(P(x) = x^{3} + 5x^{2} + 8x + 4\)

Key Concepts

Zeros of Polynomials

In polynomial functions, zeros (also called roots) are values of the variable that make the polynomial equal to zero. For example, if you have a polynomial function \( P(x) \) and \( P(c) = 0 \), then \( c \) is a zero of the polynomial. These zeros are pivotal as they give you the x-intercepts of the polynomial graph. For instance, with zeros of \(-2\) and \(-1\), as given in the exercise, these represent points where the polynomial crosses or touches the x-axis.
Identifying zeros helps in forming the factors of the polynomial, which is valuable for constructing the entire polynomial function. Each zero tells you where the graph hits the x-axis, helping visualize the function's behavior at particular points.

Multiplicity of Zeros

Multiplicity of a zero refers to how many times that particular zero appears as a solution of the polynomial equation. In simple terms, it indicates the number of times a specific zero is a root of the polynomial.
For example, if zero \(-2\) has a multiplicity of 2, it implies \((x+2)^2\) is a factor of the polynomial, meaning \(-2\) is a zero that affects the graph with a "bounce" at \(x = -2\) rather than merely passing through. The graph of the polynomial at this point will touch the x-axis and change direction.

• A multiplicity of 1 indicates that the graph crosses the x-axis.
• A higher multiplicity means the graph "bounces" off the axis.

Understanding multiplicity is crucial because it gives insights into the shape and turning points of the polynomial graph.

Constructing Polynomials

Constructing polynomials involves creating a polynomial function by using its provided zeros and their multiplicities. This is done by framing each zero and multiplicity into the form \((x - \,\text{zero})^{\text{multiplicity}}\), resulting in factors that multiply together to form the polynomial.
For example, given zeros \(-2\) with multiplicity 2, and \(-1\) with multiplicity 1, construct the polynomial by forming the factors:

• \((x + 2)^{2}\) for zero \(-2\) with multiplicity 2,
• \((x + 1)\) for zero \(-1\) with multiplicity 1.

Combining these gives the polynomial as \(P(x) = (x+2)^{2} \times (x+1)\). This multiplication process expands into the specific polynomial expression, providing a complete picture of the polynomial's form.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial, reflecting its overall behavior and complexity. It indicates the maximum number of zeros the polynomial can have and determines the end behavior of the function's graph.
For instance, a polynomial of degree 3, like the one constructed in the exercise, implies the function may have up to 3 roots, which align well with the zeros and their multiplicities given: two zeros from \(-2\) and one from \(-1\).
Recognizing the degree helps in:

• Determining the number of maximum turns the graph can make (degree - 1).
• Predicting the end behavior by indicating whether the graph approaches positive or negative infinity as \(x\) approaches positive or negative infinity.

Knowing the degree of a polynomial is essential for understanding its complexity and potential graph shape.