Problem 30 - College Algebra

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=(x+2)^{3}(x-3)^{2} $$

Step-by-Step Solution

Answer

The zeros are \(x = -2\) with multiplicity 3 and \(x = 3\) with multiplicity 2.

Step 1: Identify the function's factors

The given function is \( f(x) = (x+2)^3(x-3)^2 \). This polynomial is already factored, and we have two factors: \((x+2)^3\) and \((x-3)^2\).

Step 2: Find the zeros of each factor

To find the zeros, set each factor equal to zero. For \(x+2 = 0\), we solve to find \(x = -2\). For \(x-3 = 0\), we solve to find \(x = 3\).

Step 3: Determine the multiplicity of each zero

Multiplicity is the exponent of each factor in the factored form. The zero \(x = -2\) comes from the factor \((x+2)^3\), so its multiplicity is 3. The zero \(x = 3\) comes from the factor \((x-3)^2\), so its multiplicity is 2.

Key Concepts

Zero of a Function

When we talk about polynomials, the **zero of a function** refers to the values of variable that make the function equal to zero. In simpler terms, if you have a function \( f(x) \) and you want to find **where it crosses the x-axis**, these points are called zeros. To identify them, you set the entire polynomial function equal to zero and solve for \( x \).
For example, with the function \( f(x) = (x+2)^3(x-3)^2 \), we identify zeros by setting each distinct factor equal to zero:

• For \( (x + 2) = 0 \), solving gives \( x = -2 \) as a zero.
• For \( (x - 3) = 0 \), solving gives \( x = 3 \) as another zero.

**Zeros** are crucial in understanding the behavior of polynomial graphs, especially where the graph touches or crosses the x-axis.

Multiplicity of Roots

The concept of **multiplicity** indicates how many times a particular zero appears in a polynomial function. It is represented by the exponent of the factor associated with that zero. Higher multiplicity can affect how the graph behaves at that point.
In our example, the polynomial \( f(x) = (x+2)^3(x-3)^2 \), the zeros are identified as \( x = -2 \) and \( x = 3 \). Here’s how the multiplicity plays out:

• The zero \( x = -2 \) comes from the factor \( (x+2)^3 \), indicating a **multiplicity of 3**. The graph is tangent to the x-axis at this point—meaning it touches but does not cross the axis.
• The zero \( x = 3 \) comes from the factor \( (x-3)^2 \), indicating a **multiplicity of 2**. The graph will just "bounce" off the x-axis at this point.

This attribute of multiplicity provides insights on how the graph behaves near zeros, whether it crosses the x-axis or merely touches and turns away.

Factoring Polynomials

Factoring a polynomial involves breaking down the function into products of simpler polynomial factors. This process is very useful for identifying zeros and understanding the graph's overall shape, as each factor gives us a direct clue about the solution and behavior of the function.
Looking at our polynomial \( f(x) = (x+2)^3(x-3)^2 \), notice it is already given in its **factored form**. This makes it easier to identify zeros and their multiplicities straight away without the need for further calculations.
When factoring polynomials, the goal is to express the original polynomial as the product of linear or other polynomials. Usually, this can involve:

• Identifying and factoring out common factors.
• Using formulas for squares, cubes, and special products.
• Applying techniques like synthetic division when needed.

This process simplifies the analysis of polynomials, making it efficient to determine critical points such as zeros and their multiplicities.