Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x^{3}(x-1)^{3}(x+2) $$

Step-by-Step Solution

Answer

Zeros: 0 (multiplicity 3), 1 (multiplicity 3), -2 (multiplicity 1).

Step 1: Identify the Factors of the Polynomial

The given polynomial is \( f(x) = x^3(x-1)^3(x+2) \). It is already factored into its simplest form, where its factors are \( x^3 \), \( (x-1)^3 \), and \( (x+2) \).

Step 2: Determine the Zeros of the Polynomial

To find the zeros, set each factor equal to zero and solve for \( x \). We have the equations: \( x^3 = 0 \), \((x-1)^3 = 0 \), and \( x+2 = 0 \). Solving these gives \( x = 0 \), \( x = 1 \), and \( x = -2 \).

Step 3: Determine the Multiplicity of Each Zero

The multiplicity of a zero is determined by the power of its corresponding factor in the factored polynomial. For \( x = 0 \), the factor is \( x^3 \), so its multiplicity is 3. For \( x = 1 \), the factor is \( (x-1)^3 \), so its multiplicity is 3. For \( x = -2 \), the factor is \( x+2 \), so its multiplicity is 1.

Key Concepts

Multiplicity of Zeros

The concept of multiplicity is crucial when dealing with polynomial zeros. Not all zeros or roots are created equal. Understanding their multiplicity will help you grasp how the graph of a polynomial behaves at these points. Multiplicity refers to how many times a particular zero appears as a solution to the polynomial equation when factored completely.

When a zero has a higher multiplicity, this affects the shape of the graph at that point:

• If a zero has a multiplicity of 1, the graph will cross the x-axis at this point.
• If a zero has an even multiplicity (like 2, 4, etc.), the graph will only touch the x-axis at this point and bounce back.
• If a zero has an odd multiplicity greater than 1 (like 3, 5, etc.), the graph will cross the x-axis but flatten out somewhat at the zero.

This knowledge helps you not only to find the zeros but also to understand the general shape of the polynomial graph.

Factored Polynomials

Factored polynomials are expressions that are broken down into products of simpler polynomials. Factoring is often one of the first steps in solving polynomial equations because it simplifies the process of finding zeros or roots.

In the example polynomial given, \[ f(x) = x^3(x-1)^3(x+2) \], we see it's already in factored form:

• \( x^3 \) is a factor corresponding to a zero at \( x = 0 \).
• \((x-1)^3\) is a factor corresponding to a zero at \( x = 1 \).
• \(x+2\) is a factor corresponding to a zero at \( x = -2 \).

Each factor provides direct insight into the zeros of the polynomial and their respective multiplicities. The beauty of factored form is its ability to clearly show the individual components of a polynomial and simplify solving for zeros.

Finding Zeros of a Polynomial

Finding the zeros of a polynomial is a foundational skill in algebra. It involves determining the values of \( x \) for which the polynomial equals zero. The zeros, sometimes called roots or solutions, are the \( x \)-values where the polynomial's graph intersects the x-axis.

To find the zeros of any polynomial, especially when it is in factored form like \[ f(x) = x^3(x-1)^3(x+2) \], follow these steps:

• Set each factor of the polynomial equal to zero.
• Solve the resulting simple equations.

For example:

• Setting \( x^3 = 0 \), we find \( x = 0 \).
• Setting \( (x-1)^3 = 0 \), we find \( x = 1 \).
• Setting \( x+2 = 0 \), we find \( x = -2 \).

Each solution represents a zero of the original polynomial, and knowing their multiplicity allows us to understand how the graph behaves at these points.