Problem 252 - Precalculus

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: \(x = -2\) (multiplicity 3), \(x = 3\) (multiplicity 2).

Step 1: Identify the Zeros

To find the zeros of the function \( f(x) = (x+2)^3 (x-3)^2 \), we need to determine the values of \( x \) that make the entire expression equal to zero. This happens when either \((x+2)^3 = 0\) or \((x-3)^2 = 0\).

Step 2: Solve for Zero Conditions

First, solve \((x+2)^3 = 0\). This gives \(x+2 = 0\), so \(x = -2\).\ Next, solve \((x-3)^2 = 0\). This gives \(x-3 = 0\), so \(x = 3\).

Step 3: Determine Multiplicities

The zero \(x = -2\) is found from the factor \((x+2)^3\), so it has a multiplicity of 3.The zero \(x = 3\) is found from the factor \((x-3)^2\), so it has a multiplicity of 2.

Key Concepts

Zeros of Polynomials

In mathematics, the zeros of a polynomial are the values of \( x \) for which the polynomial equals zero. Finding these zeros is an essential part of understanding the behavior of polynomial functions. When a polynomial function is equal to zero, it means that the graph of the polynomial intersects the x-axis at these points. Therefore, zeros are also called x-intercepts.
To determine these zeros, you need to look for values of \( x \) that make any factor of the polynomial zero. For example, in the function \( f(x) = (x+2)^3 (x-3)^2 \), we set each factor of the polynomial to zero:

• \( (x+2)^3 = 0 \) gives \( x = -2 \)
• \( (x-3)^2 = 0 \) gives \( x = 3 \)

Identifying these values helps you understand where the polynomial touches or crosses the x-axis.
More complicated polynomials may require additional algebraic techniques or numerical methods, but the foundational idea remains the same: solve for when the polynomial equals zero.

Multiplicity of Roots

In polynomials, the concept of multiplicity refers to the number of times a particular root is repeated. If a polynomial has a root that appears more than once, this is called a repeated root. Each occurrence of a root corresponds to a factor's exponent.
For instance, in the polynomial \( f(x) = (x+2)^3 (x-3)^2 \), we have:

• The root \( x = -2 \) comes from the factor \( (x+2)^3 \), indicating it is a repeated root with a multiplicity of 3.
• Similarly, the root \( x = 3 \) is derived from the factor \( (x-3)^2 \), showing it has a multiplicity of 2.

Multiplicity influences the shape of a polynomial's graph at the x-intercepts:

• A root with an odd multiplicity, such as 1 or 3, means that the graph will cross the x-axis at this intercept.
• A root with an even multiplicity, like 2, implies that the graph merely touches the x-axis and turns around at this intercept.

Recognizing multiplicities helps in sketching the graph and understanding the polynomial's behavior near its zeros.

Factored Form of Polynomials

The factored form of a polynomial expresses it as a product of its linear factors, each possibly raised to a power. This form is useful for finding roots and understanding the polynomial's behavior.
Given a polynomial like \( f(x) = (x+2)^3 (x-3)^2 \), the factored form is already provided. Each factor corresponds directly to a zero of the polynomial, and the exponent of each factor gives the multiplicity of the corresponding root.
The factored form:

• makes it straightforward to identify zeros and their multiplicities.
• helps in decomposing complex polynomial expressions into simpler parts.

For students, understanding and working with factored forms allows for easier manipulation and solution-finding, especially when dealing with higher-degree polynomials. Factored form is not only a step in solving equations but also a tool for deepening comprehension of polynomial functions.
When creating the factored form, look for common factors and roots using methods like factoring by grouping, synthetic division, or trial and error with possible rational roots.