Problem 16 - Precalculus

Question

List the distinct roots of each equation. In the case of a repeated root, specify its multiplicity. $$x(x+5)^{4}=0$$

Step-by-Step Solution

Answer

Roots: \(x = 0\) (multiplicity 1) and \(x = -5\) (multiplicity 4).

Step 1: Identify the Equation Parts

The given equation is \(x(x+5)^4 = 0\). This is a product of terms set equal to zero, indicating it's a polynomial equation. We recognize that the product of these terms equals zero when any individual term is zero.

Step 2: Apply the Zero Product Property

According to the zero product property, if a product of terms is equal to zero, at least one of the factors must be zero. So, we set each factor equal to zero: \(x = 0\) and \((x + 5)^4 = 0\).

Step 3: Find the Roots

First, solve \(x = 0\). This gives the root \(x = 0\). Second, solve \((x + 5)^4 = 0\). A polynomial \((x + 5)^n\) equals zero when \(x + 5 = 0\) (regardless of \(n\)), giving the root \(x = -5\).

Step 4: Determine Multiplicities

The root \(x = 0\) comes from the term \(x\) in the polynomial and appears once, so its multiplicity is 1. The root \(x = -5\) arises from the term \((x+5)^4\), which means it appears four times, giving it a multiplicity of 4.

Step 5: List the Roots and Their Multiplicities

The distinct roots of the equation are \(x = 0\) with a multiplicity of 1 and \(x = -5\) with a multiplicity of 4.

Key Concepts

Zero Product Property

Every time you encounter an equation that looks like a bunch of multiplied terms equaling zero, you're in luck! The Zero Product Property is here to simplify your life. This property says that if you multiply several terms together and end up with zero, then at least one of those terms must be zero. Remember:

• When you see an equation like \( x(x+5)^4 = 0 \), think of it as a bunch of friends holding hands. If their holding hands causes them to fall (a.k.a equals zero), then at least one of them (one factor) is on the ground (zero).
• This means you can just go through and set each factor equal to zero, like taking a roll call: first \(x = 0\), then \((x+5)^4 = 0\).

This helps us break down potentially complex equations into simpler parts, which is a powerful tool in your math toolkit.

Multiplicity of Roots

When dealing with polynomials, certain roots might show up more than once. This is where the term "multiplicity" comes into play. For instance:

• The equation \((x + 5)^4 = 0\) gives us the root \(x = -5\).
• But, notice the exponent of 4 in \((x+5)^4\). This signals something important: multiplicity!

In this context, the root \(x = -5\) isn't just a single occurrence – it appears four times over. We say that its multiplicity is 4. On the other hand, the root derived from \(x = 0\) only shows up once, giving it a multiplicity of 1. Recognizing the multiplicity of roots helps us understand the behavior and nature of the graph of a polynomial function. Repeated roots imply that the graph will just "touch" the x-axis and turn around, instead of passing through it.

Polynomial Equation

A polynomial equation is an expression that involves sums, products, and powers of variables. In our given exercise, the polynomial equation is \(x(x+5)^4 = 0\). Here's what makes it special:

• It consists of a variable \(x\) raised to various powers, particularly higher powers and terms multiplied together.
• Polynomials are like the building blocks of algebra, showing up in equations of all sorts of complexity.
• The focus is often on finding the roots, which are values of \(x\) that make the entire expression equal to zero.

Understanding that polynomial equations can be broken down into factors and solved using properties like the zero product property provides an essential foundation for tackling more complex mathematical challenges. They are everywhere in math and science and being familiar with them is key to becoming adept in these areas.