Problem 20 - Precalculus

Question

Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is 7 or more, express \(f(x)\) in factored form; otherwise, express \(f(x)\) in the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\). $$\begin{array}{lll} \hline \text { Root } & 0 & 4 \\ \text { Multiplicity } & 2 & 1 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The polynomial is \(f(x) = x^3 - 4x^2\).

Step 1: Identifying Roots and Multiplicities

The given roots and their multiplicities are as follows: 0 with multiplicity 2 and 4 with multiplicity 1.

Step 2: Constructing Factors from Roots

To find the polynomial, construct factors corresponding to each root. For the root 0 with multiplicity 2, we have the factor \((x-0)^2 = x^2\). For the root 4 with multiplicity 1, we have the factor \((x-4)^1 = x-4\).

Step 3: Formulating the Polynomial

Using the factors from Step 2, we multiply them to form the polynomial: \(f(x) = (x^2)(x-4)\).

Step 4: Expanding the Polynomial

Expand the expression \((x^2)(x-4)\) to express the polynomial in standard form: \[f(x) = x^2(x-4) = x^3 - 4x^2\].

Step 5: Ensuring Leading Coefficient is 1

Check the polynomial \(x^3 - 4x^2\). The leading coefficient is 1, which satisfies the problem requirement.

Key Concepts

Roots and Multiplicities

In understanding polynomials, the concept of roots and their multiplicities is central. A root of a polynomial is a solution to the equation where the polynomial equals zero, that is, \(f(x) = 0\). The multiplicity of a root refers to how many times that root appears in the factored form of the polynomial.
For instance, if a polynomial has a root of zero with multiplicity 2, it means the factor \(x-0 = x\) appears twice in the factored polynomial, \(x^2\). This is like saying the number zero is a root of the polynomial but has a weight or strength of 2. If another root, say 4, appears with multiplicity 1, it simply means it appears once as a factor, \(x-4\).
The concept is important because it helps in identifying the behavior of polynomial graphs. At a root with odd multiplicity, the graph will cross the x-axis, whereas with even multiplicity, it will touch the axis and turn back.

Factored Form

The factored form of a polynomial is a powerful way to express all its roots clearly and concisely. This form involves breaking down the polynomial into a product of its factors, each linked with a root by the equation \(x-r_k\), where \(r_k\) is a specific root.
For example, in the exercise given, we utilized the roots and their multiplicities to construct the factors \(x^2\) for the root zero, and \(x-4\) for the root four. Thus, the polynomial expression in factored form is \(f(x) = (x^2)(x-4)\).
Factored forms make it clear where the roots of the polynomial lie and how they are repeated. Additionally, for higher degree polynomials (degree 7 or more), maintaining this form aids in simplifying complex calculations.

Polynomial Expansion

Polynomial expansion refers to the process of multiplying out the factors in a factored form polynomial to express it in standard form. Standard form is where the polynomial is expanded and presented as a sum of terms with decreasing powers of \(x\).
In our exercise, polynomial expansion involves multiplying the factors \( (x^2)(x-4)\) to get a simplified structure: \( x^3 - 4x^2 \). Expanding simplifies the visual representation and allows us to see each term and its respective coefficient.
The standard form of a polynomial is also important because it makes it easier to identify features like the degree of the polynomial and the leading coefficient. Here, after expansion, we confirmed that the leading coefficient was 1, fulfilling the problem's conditions. Polynomial expansion thus serves to both simplify and verify the requirements of the problem at hand.