Problem 16 - Precalculus: Functions and Graphs

Question

Find the zeros of \(f(x),\) and state the multiplicity of each zero. $$f(x)=x(x+1)^{4}(3 x-7)^{2}$$

Step-by-Step Solution

Answer

Zeros: 0 (mult. 1), -1 (mult. 4), 7/3 (mult. 2).

Step 1: Identify the Factors of the Function

The given function is \(f(x) = x(x+1)^4(3x-7)^2\). To find the zeros, we need to identify each of the factors that make up this polynomial function.

Step 2: Set Each Factor to Zero

The zeros of the function occur when any of the factors are equal to zero. So we set each factor equal to zero: \(x = 0\), \((x+1) = 0\), and \((3x-7) = 0\).

Step 3: Solve for Each Zero

- Solving \(x = 0\) gives a zero at \(x = 0\).- Solving \(x+1 = 0\) gives \(x = -1\).- Solving \(3x-7 = 0\) gives \(x = \frac{7}{3}\).

Step 4: Determine the Multiplicity of Each Zero

Multiplicity refers to the power to which each factor is raised. For \(x = 0\), the factor \(x\) is raised to the first power, so its multiplicity is 1. For \(x = -1\), since \((x+1)\) is raised to the fourth power, the multiplicity of \(x = -1\) is 4. For \(x = \frac{7}{3}\), since \((3x-7)\) is raised to the second power, the multiplicity is 2.

Key Concepts

Zeros of a polynomial

In mathematics, the zeros of a polynomial are the values of the variable that make the entire polynomial equal zero. Simply put, these are the solutions to the equation formed by setting the polynomial equal to zero.
For example, if you have a polynomial function like \(f(x) = x(x+1)^4(3x-7)^2\), finding the zeros involves determining the values of \(x\) that satisfy the equation \(f(x) = 0\). Here is how it breaks down:

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

This gives us the zeros of the polynomial:

• \(x = 0\)
• \(x = -1\)
• \(x = \frac{7}{3}\)

Identifying the zeros of a polynomial is crucial because they provide insights into the graph of the function and its behavior along the x-axis. Whenever a polynomial crosses or touches the x-axis, it does so at its zeros.

Multiplicity of zeros

The concept of multiplicity of zeros refers to how many times a particular zero appears as a solution of the equation. In the context of polynomial functions, observing the multiplicity can tell us more about the behavior of the polynomial around those zeros.
Each factor of a polynomial can be raised to a power, which indicates its multiplicity:

• If a zero occurs once, its multiplicity is 1. For example, in the polynomial \(x(x+1)^4(3x-7)^2\), the factor \(x\) results in zero \(x = 0\) with multiplicity 1.
• If a zero occurs more than once, its multiplicity is greater than 1. For \(x+1=0\), we solve to get \(x=-1\) and it has a multiplicity of 4. Similarly, \(3x-7=0\) gives \(x=\frac{7}{3}\) with multiplicity 2.

Understanding multiplicity is valuable because it not only shows the frequency of each zero but also determines how the graph of a polynomial touches or crosses the x-axis at these points:

• If the multiplicity is odd, the graph crosses the x-axis at the zero.
• If the multiplicity is even, the graph touches and bounces off the x-axis.

In the given function, \(x = 0\) crosses, \(x = -1\) touches and bounces off, and \(x = \frac{7}{3}\) touches and bounces off the x-axis.

Factoring polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This is a major step in finding the zeros of the polynomial because these factors correspond to the equations that we set to zero to solve for the variable.
The process of factoring can include several techniques:

• Identifying common factors
• Grouping terms
• Using the difference of squares
• Applying the quadratic formula when necessary

For instance, consider the polynomial \(f(x) = x(x+1)^4(3x-7)^2\). It is already factored into its simplest components. We see:

• The factor \(x\) naturally means \(x\) itself is a zero.
• The factor \((x+1)^4\) signifies the repeated zero \(x = -1\).
• The factor \((3x-7)^2\) gives us the zero \(x = \frac{7}{3}\).

This form makes it straightforward to set each factor to zero and solve for the variable. Factoring is a powerful tool because it breaks down polynomials into simpler, manageable parts, allowing us to easily find solutions and understand the behavior of polynomial expressions.