Problem 7 - Algebra for College Students

Question

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{5}-4 x^{3}=0$$

Step-by-Step Solution

Answer

The degree is 5. The roots are 0 (multiplicity 3), 2 (multiplicity 1), and -2 (multiplicity 1).

Step 1: Identify the Degree of the Polynomial

Look at the highest power of the variable in the polynomial equation. The degree of the polynomial is the highest exponent of the variable. For the equation \(x^{5}-4x^{3}=0\), the highest power of \(x\) is 5. Therefore, the degree of the polynomial is 5.

Step 2: Factor the Polynomial Equation

First, factor out the greatest common factor from the polynomial. In this case, the greatest common factor is \(x^3\). So, \(x^{5}-4x^{3}=0\) can be factored as \(x^3(x^2-4)=0\).

Step 3: Solve for the Roots

Set each factor equal to zero and solve for \(x\). For \(x^3 = 0\), the solution is \(x = 0\). For \(x^2 - 4 = 0\), rewrite it as \(x^2 - 2^2 = 0\), which factors further into \((x-2)(x+2) = 0\). Solving these, we get \(x = 2\) and \(x = -2\).

Step 4: Determine the Multiplicity of Each Root

Multiplicity refers to the number of times a particular root occurs. The root \(x = 0\) comes from \(x^3\) and occurs 3 times. The roots \(x = 2\) and \(x = -2\) each occur once. Thus, \(x = 0\) has a multiplicity of 3, while \(x = 2\) and \(x = -2\) each have a multiplicity of 1.

Key Concepts

Degree of Polynomial

The degree of a polynomial is a critical concept in understanding its behavior and properties. The degree is determined by the highest exponent of the variable in the polynomial. For instance, in the given polynomial equation, \(x^{5}-4x^{3}=0\), the term with the highest power of \(x\) is \(x^{5}\). Therefore, the degree of this polynomial is 5. Knowing the degree can help you predict the number and nature of the polynomial's roots.

Factoring Polynomials

Factoring polynomials is a method used to break down a polynomial into simpler components called factors. These factors, when multiplied together, give back the original polynomial. In the polynomial \(x^{5}-4x^{3}=0\), we start by factoring out the greatest common factor (GCF). Here, the GCF is \(x^{3}\), giving us:

\(x^{3}(x^{2} - 4) = 0\).
Next, we recognize that \(x^{2} - 4\) is a difference of squares, which can be factored further into:
\((x-2)(x+2)\).
Thus, the fully factored form of the polynomial is:
\(x^{3}(x-2)(x+2) = 0\).This factorization simplifies the process of finding the polynomial's roots.

Root Multiplicity

The concept of root multiplicity refers to the number of times a particular root occurs in a polynomial. For the polynomial \(x^3 (x-2)(x+2) = 0\), the roots can be found by setting each factor equal to zero. This gives us the roots:

• \(x = 0\)
• \(x = 2\)
• \(x = -2\)

The multiplicity of a root is determined by the exponent of the factor. For \(x^3 = 0\), the root \(x = 0\) occurs 3 times, giving it a multiplicity of 3. The roots \(x = 2\) and \(x = -2\) each occur once in the factors \((x-2)\) and \((x+2)\), so their multiplicity is 1. Recognizing multiplicity is important for understanding the polynomial's graph and its intersection points with the x-axis.

Solving Polynomial Equations

To solve polynomial equations, we typically follow a sequence of steps:
1. **Identify the Degree**: Determine the highest exponent to understand the number of potential roots.
2. **Factor the Polynomial**: Break down the polynomial into simpler factors to easily find roots.
3. **Solve for the Roots**: Set each factor equal to zero and solve for the variable.
For the polynomial \(x^{5}-4x^{3}=0\), factoring gave us \(x^{3}(x-2)(x+2) = 0\).
By setting each factor to zero, we get the roots:
\(x = 0\) (with multiplicity 3), \(x = 2\), and \(x = -2\).
These steps simplify the process of finding roots and understanding the polynomial's properties. Solving polynomial equations is essential in various mathematical and real-world applications.