Problem 45 - 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 -2 and \(-1,\) each of multiplicity 1 zero at 5 of multiplicity 3

Step-by-Step Solution

Answer

The polynomial function with the specified properties is \(f(x) = (x + 2) \cdot (x + 1) \cdot (x - 5)^3\). It should be noted, though, that other formulas may also match these properties, provided that the equation has the same roots with the same multiplicities and is of degree 5.

Step 1: Identify the roots and their multiplicities

From the given properties of the polynomial, we can deduce that the roots of the polynomial are \(x = -2, -1, 5\) with multiplicities 1, 1, and 3 respectively.

Step 2: Construct the polynomial function

We can formulate the polynomial function utilising the noted information. Using the formula \(f(x) = a \cdot (x - r1)^{k1} \cdot ...(x - rn)^{kn}\), we can design the function with a for normalization, not changing the roots or multiplicities. So the function can be given as \(f(x) = a \cdot (x - (-2))^1 \cdot (x - (-1))^1 \cdot (x - 5)^3\). Simplifying this gives \(f(x) = a \cdot (x + 2) \cdot (x + 1) \cdot (x - 5)^3\).

Step 3: Conclude the polynomial function

Choose \(a=1\) for the simplest expression. Therefore, our final function will be \(f(x) = (x + 2) \cdot (x + 1) \cdot (x - 5)^3\).

Key Concepts

Polynomial Roots

Understanding polynomial roots is crucial in mathematics, especially when examining the behavior of polynomial functions. A root of a polynomial is a solution to the equation formed when the polynomial function is set equal to zero, that is, it's the value of 'x' for which the polynomial evaluates to zero. For instance, in the polynomial function \(f(x) = x^2 - 5x + 6\), the roots can be found by solving the equation \(x^2 - 5x + 6 = 0\).

Roots can be real or complex numbers and are sometimes referred to as zeroes of the polynomial since they 'zero out' the function. To find the roots of a given polynomial, it's often helpful to factor the equation if possible, or otherwise apply methods such as the quadratic formula, synthetic division, or numerical methods for higher degree polynomials. The roots of a polynomial are tied intimately to its graph; each real root corresponds to a point where the graph intersects the x-axis.

Root Multiplicity

When dealing with polynomial roots, root multiplicity refers to the number of times a particular root occurs. A multiple root, also known as a repeated root, is a root that has a multiplicity greater than 1. For example, let's consider the polynomial \(f(x) = (x - 3)^2(x + 4)\). Here, the root \(x=3\) has a multiplicity of 2 because it appears twice in the factored form (squared), whereas \(x=-4\) has a multiplicity of 1.

The multiplicity of a root affects the graph of the polynomial. If a root's multiplicity is odd, the graph will cross the x-axis at that point. If it's even, the graph touches and rebounds from the x-axis at the root's location. High multiplicity can also indicate a 'flatter' behavior of the graph around that root. This concept is particularly important when sketching the plot of a polynomial function or predicting its behavior without a graph.

Constructing Polynomial Functions

The process of constructing polynomial functions given specific roots and multiplicities is a methodical one. To construct such a function, start by converting each root and its respective multiplicity into a factorial term of the form \( (x - r)^m \), where \(r\) is the root, and \(m\) its multiplicity. Assemble these terms to form the polynomial. In the exercise provided, we used this technique to build a polynomial of degree 5.

Here are the detailed steps we followed:

Starting with the Roots

Begin by writing a factor for each root, incorporating its multiplicity. For the given roots \(x = -2\), \(x = -1\), and \(x = 5\) with multiplicities 1, 1, and 3, we write \( (x+2)^1 \), \( (x+1)^1 \), and \( (x-5)^3 \).

Forming the Function

By multiplying these factors, we construct the function: \( f(x) = a \times (x+2) \times (x+1) \times (x-5)^3 \). The coefficient \('a'\) is a scaling factor that doesn't change the roots or their multiplicities but can affect the width and orientation of the graph.

With the roots and multiplicities accounted for, you can choose a suitable value for \(a\) to fulfill any additional criteria. In the context of our exercise, setting \(a = 1\) yielded the simplest form of the function, and it maintains the behavior characteristic of a fifth-degree polynomial. This approach is fundamental in understanding how polynomial functions work and creates a bridge between algebraic expressions and their graphical representations.