Problem 30 - College Algebra with Corequisite Support

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

Zeros are \(x = -2\) (multiplicity 3) and \(x = 3\) (multiplicity 2).

Step 1: Identify the Factors

The given function is \( f(x) = (x+2)^{3}(x-3)^{2} \). This expression is already factored, making it easier to identify the zeros.

Step 2: Set the Factors to Zero

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

Step 3: Solve for the Zeros

Solve each equation from the factors set to zero: \((x+2) = 0\) leads to \(x = -2\), and \((x-3) = 0\) leads to \(x = 3\).

Step 4: Identify the Multiplicities

The multiplicity is the power to which each factor is raised. For \(x = -2\), the factor is \((x+2)\) raised to the 3rd power, so the multiplicity is 3. For \(x = 3\), the factor is \((x-3)\) raised to the 2nd power, so the multiplicity is 2.

Key Concepts

Multiplicity of Zeros

Understanding the multiplicity of zeros is an important part of polynomial equations. The multiplicity of a zero refers to the number of times a particular zero is a root of a polynomial equation.
For example, if a polynomial has a zero at a certain point and the associated factor appears multiple times, its multiplicity is more than 1.
In the context of the polynomial \( f(x) = (x+2)^{3}(x-3)^{2} \):

• The zero \( x = -2 \) with the factor \( (x+2) \) appears three times, indicating the multiplicity of 3.
• Similarly, the zero \( x = 3 \) with the factor \( (x-3) \) appears twice, so its multiplicity is 2.

Identifying multiplicities helps to understand the behavior of the graph of the polynomial. If a zero has an even multiplicity, the graph touches and turns at that x-value but doesn’t cross the x-axis. If it’s odd, the graph crosses the x-axis. This insight provides a deeper comprehension of how the polynomial behaves.

Factored Form of Polynomials

Polynomials can often be expressed in factored form to reveal their zeros directly and make solving them easier. Factored form displays a polynomial as a product of its linear factors raised to respective powers.
For example, the polynomial \( f(x) = (x+2)^{3}(x-3)^{2} \) is already in its factored form. Here, each set of parentheses represents a factor of the equation, and the exponent indicates how many times that factor occurs.
Advantages of placing a polynomial in factored form include:

• Easier identification of zeros, as each factor can be set to zero to solve for the roots.
• It helps to find the multiplicity of those zeros based on the exponent of each factor.

Being familiar with factored forms simplifies solving and graphing polynomial functions. Factored forms provide clear information about potential turning points and the overall shape of the polynomial's graph.

Solving Polynomial Equations

Solving polynomial equations often begins with expressing them in factored form, whenever possible. This strategy simplifies the process of finding zeros, as each factor corresponds to a potential solution given by setting it equal to zero.
To solve the polynomial \( f(x) = (x+2)^{3}(x-3)^{2} \):

• First, recognize that the polynomial is already factored, with factors \((x+2)\) and \((x-3)\).
• Next, set each factor equal to zero: \( (x+2)=0 \) and \( (x-3)=0 \).
• Solve each equation to find \( x = -2 \) and \( x = 3 \), which are the zeros of the polynomial.

Understanding this process paves the way for analyzing any polynomial equation's structure and predicting how its graph behaves. Using factored forms not only identifies zeros but also provides insight into the polynomial function's multiplicity and vertex crossings.