Problem 77 - 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: \(-4,\) multiplicity: 2 Zero: \(3,\) multiplicity: 2 Degree: 4

Step-by-Step Solution

Answer

The polynomial function with zeros -4 and 3 each of multiplicity 2 and of degree 4 is \(f(x) = x^4 - 4x^3 -20x^2 +176x + 144\).

Step 1: Find the factors from the zeros

We are given the zeros -4 and 3, each with a multiplicity of 2. So the factors of the polynomial would be \((x -(-4))^2\) and \((x -3)^2\). This simplifies to \((x + 4)^2\) and \((x -3)^2\).

Step 2: Combine the factors to form the polynomial

Now, the factors are multiplied together to form the polynomial function. So the polynomial function \(f(x)\) is given by \(f(x) = (x + 4)^2 * (x - 3)^2\). This further simplifies to \(f(x) = (x^2 + 8x + 16) * (x^2 - 6x + 9)\).

Step 3: Expand and Simplify the Polynomial Function

To obtain the polynomial in standard form, we multiply the two quadratics and simplify the result. Performing the multiplication and combining like terms, we get \(f(x) = x^4 - 4x^3 -20x^2 +176x + 144\).

Key Concepts

Polynomial Roots

Understanding polynomial roots is essential to grasp the foundation of solving polynomial equations and constructing polynomial functions. Roots, also known as zeros, are the solutions to the polynomial equation where the function equals zero. When we say a number is a 'root' of a polynomial, it means that if you plug this number into the polynomial, the outcome is zero.

For instance, if you are given a polynomial function like f(x) = x^2 - 5x + 6, and you want to find its roots, you are looking for values of 'x' that make f(x) = 0. Factorizing the quadratic equation, we get (x - 2)(x - 3) = 0, which implies the roots are x = 2 and x = 3. These are the values that, when substituted back into the polynomial, yield a result of zero. In the context of an exercise, finding the roots can guide us to reconstruct the original polynomial, as it suggests the factors and thus the behavior of the graph at these points.

Multiplicity of Zeros

The multiplicity of a root is a whole number representing the number of times the root occurs as a solution of the polynomial equation. A root's multiplicity affects the shape of the polynomial's graph at that x-value. When a polynomial has a zero with multiplicity 1, the graph crosses the x-axis at that point. However, if the multiplicity is even, the graph touches the axis and turns back without crossing it.

A prime example of this concept would be the polynomial f(x)=(x+4)^2 which has the root -4 with multiplicity 2. This means at x=-4, the graph of the function touches the x-axis and curves back upwards, creating a parabolic shape. Multiplicities higher than 1 result in flatter touches or bounces at the root. Understanding multiplicity is crucial for graphing polynomials and predicting the exact behavior at each zero because it sets the foundation for the next step, which is constructing polynomials.

Constructing Polynomials

Constructing polynomials from given roots and multiplicities is a bit like assembling a puzzle - each piece must fit perfectly to create the intended image, or in our case, the polynomial function. Once you know the zeros of a polynomial and their respective multiplicities, you can work backward to create a polynomial equation. To construct a polynomial function, follow these simplified steps:

Step 1: Translate Roots into Factors

Use each zero to create a factor of the polynomial. If you have a zero at x=a, the factor will be (x-a). Repeat this for each zero.

Step 2: Apply the Multiplicity

Each factor corresponding to a root is raised to the power of its multiplicity. So if the root x=a has a multiplicity of m, the factor becomes (x-a)^m.

Step 3: Multiply the Factors Together

Combine the individual factors you've created by multiplying them to form the polynomial. If the polynomial is meant to have a specific degree, make sure that the resulting polynomial's degree matches it.

Step 4: Expand and Simplify

Once all the factors are combined, expand them to write the polynomial in the standard form, which is a single expression in descending powers of x.

For instance, with zeros -4 and 3, both having a multiplicity of 2, and desiring to construct a fourth-degree polynomial, we'd start with factors (x+4)^2 and (x-3)^2. Multiplying these factors and then expanding them gives us the final polynomial function. It's important to verify that the polynomial equation has the correct degree and that it produces the expected zeros and behavior at those zeros. Checking your work through substitution and graphing can help confirm the accuracy of the constructed polynomial.