Problem 32 - College Algebra

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 are 0 (mult. 3), 1 (mult. 3), and -2 (mult. 1).

Step 1: Identify factors

The given function is \( f(x)=x^{3}(x-1)^{3}(x+2) \). We can observe that it is already written in factored form and the factors are \( x \), \( x-1 \), and \( x+2 \). These factors help us determine the zeros of the function.

Step 2: Find the zeros

Set each factor of the function equal to zero to find the zeros.1. For the factor \( x \), \( x=0 \).2. For \( x-1 \), set \( x-1=0 \), giving \( x=1 \).3. For \( x+2 \), set \( x+2=0 \), giving \( x=-2 \). So, the zeros are \( 0, 1, \) and \( -2 \).

Step 3: Determine the multiplicity

The multiplicity of a zero is determined by the exponent of its factor in the factorized expression:1. The zero \( x=0 \) comes from the factor \( x^3 \), so its multiplicity is 3.2. The zero \( x=1 \) comes from the factor \( (x-1)^3 \), so its multiplicity is also 3.3. The zero \( x=-2 \) comes from the factor \( (x+2) \), so its multiplicity is 1.

Key Concepts

Zeros of a Polynomial

In mathematics, zeros of a polynomial are the values of the variable that make the polynomial equal to zero. When a polynomial is expressed in its factored form, it's easier to identify these zeros. For example, with the polynomial function \( f(x) = x^3(x-1)^3(x+2) \), identifying each factor as a separate expression can help find where the polynomial equals zero. To determine the zeros, set each factor to zero and solve for \( x \). Consequently, the solutions are:

• For the factor \( x \): solving \( x = 0 \) gives the zero \( x = 0 \).
• For the factor \( x - 1 \): solving \( x - 1 = 0 \) provides \( x = 1 \).
• For the factor \( x + 2 \): solving \( x + 2 = 0 \) yields \( x = -2 \).

So, the zeros of the polynomial \( f(x) \) are \( 0, 1, \) and \( -2 \). These zeros tell us that the graph of the polynomial will intersect or touch the x-axis at these points.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears in the factored form of a polynomial. It is determined by the number of times a factor is repeated. For the polynomial \( f(x) = x^3(x-1)^3(x+2) \), each zero has an associated multiplicity:

• Zero \( x = 0 \): arises from the factor \( x^3 \), indicating this zero has a multiplicity of 3.
• Zero \( x = 1 \): derived from the factor \( (x-1)^3 \), hence this zero also has a multiplicity of 3.
• Zero \( x = -2 \): comes from the factor \( (x+2) \), giving it a multiplicity of 1.

The multiplicity of a zero affects the behavior of the polynomial's graph at the intercept. A zero with an odd multiplicity crosses the x-axis, whereas a zero with an even multiplicity only touches the x-axis without crossing it.

Factored Form

The factored form of a polynomial makes identifying zeros straightforward and is foundational for solving polynomial equations. A polynomial is expressed in factored form when written as a product of its factors. For example, in \( f(x) = x^3(x-1)^3(x+2) \), the expression is already in its factored form:

• \( x^3 \) - contributes to the zero \( x = 0 \).
• \( (x-1)^3 \) - corresponds to the zero \( x = 1 \).
• \( (x+2) \) - identifies the zero \( x = -2 \).

By observing the factored form, one can quickly find the zeros by setting each factor equal to zero. Furthermore, the exponents of the factors reveal the multiplicity of the corresponding zeros. This form provides a clear view of the polynomial's roots and their multiplicity, offering insights into the polynomial's graphical behavior.