Problem 76 - 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: \(3,\) multiplicity: 1 Zero: \(2,\) multiplicity: 3 Degree: 4

Step-by-Step Solution

Answer

The polynomial function with the given zeros, multiplicities and degree is \(f(x) = x^4 - 7x^3 + 18x^2 - 20x + 12\).

Step 1: Use the given zeros and multiplicities to construct the factors

Remember, for a factor with a multiplicity of \(m\) and a root \(r\), you can represent the factor as \((x - r)^m\). Hence, the factors for the given roots and multiplicities are: \((x - 3)^1\) and \((x - 2)^3\).

Step 2: Use the factors to form the polynomial function

Multiply all the obtained factors to derive the polynomial function. Therefore, the polynomial function becomes \(f(x) = (x - 3)(x - 2)^3\).

Step 3: Expand the equation

Simplify the function by expanding the equation to obtain the final function. Hence the simplified function becomes \(f(x) = x^4 - 7x^3 + 18x^2 - 20x + 12\).

Key Concepts

Zeros and Multiplicities

Understanding zeros and multiplicities is essential when working with polynomial functions. A zero of a polynomial is a value at which the function equals zero. Essentially, it's where the graph of the polynomial crosses or touches the x-axis. Each zero can have a multiplicity, indicating how many times it appears as a factor.

For example, in the polynomial from the exercise, we have two zeros: 3 and 2. The zero 3 has a multiplicity of 1, meaning it appears once as a factor. This makes the factor \(x - 3\). Zero 2 has a multiplicity of 3, so its factor is \(x - 2\)^3. This means the graph just touches or bounces off the x-axis at zero 2, indicating that the root is repeated.

Identifying the zeros and their multiplicities helps in forming the polynomial accurately.

Degree of a Polynomial

The degree of a polynomial is a crucial concept, as it determines the behavior and shape of the polynomial's graph. The degree is the highest exponent of the variable in the polynomial when it is expanded.

In our exercise, we are asked to find a polynomial with a degree of 4. This means, after expanding, the polynomial should have the highest power of x as 4. The degree helps determine the number of turning points and the end behavior of the graph. For example, a fourth-degree polynomial might have up to three turning points (changes in direction).

The degree also suggests the number of zeros a polynomial potentially can have, including complex and repeated zeros.

Factorization

Factorization involves breaking down a polynomial into its linear factors based on its zeros and multiplicities. Each root provides a corresponding factor of the form \(x - r\), where \(r\) is the root.

Using zeros and their multiplicities from the problem, we factored the polynomial as \(x - 3\)^1 and \(x - 2\)^3. This means the function can be written as \(f(x) = (x - 3)(x - 2)^3\). By factorizing a polynomial, you can easily determine its zeros and their behavior. It simplifies the process of solving equations or graphing.

Besides aiding with solutions, factorization is also used in simplifying and understanding polynomials further.

Expanding Polynomials

Expanding polynomials involves multiplying the factors to express the polynomial in standard form. This form shows all terms completely expanded without parenthesis, revealing the polynomial's degree and coefficients.

In the given exercise, after factorizing as \(f(x) = (x - 3)(x - 2)^3\), we expand it to get the polynomial in its standard form. The expanded polynomial becomes \(f(x) = x^4 - 7x^3 + 18x^2 - 20x + 12\).

Expanding is useful because it provides insights into the polynomial's characteristics like its degree and helps in sketching the graph. While expanding, pay close attention to multiplication rules and combine like terms for correct results. The expanded form is crucial for further analysis or solving equations.