Graph The Polynomial Function $f(x)=x(3-x)(5-x)$ Using Parts (a) Through (e).Determine The Zero(s) And Their Multiplicity. Use This Information To Determine Whether The Graph Crosses Or Touches The $x$-axis At Each

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Introduction

Polynomial functions are a fundamental concept in mathematics, and graphing them is an essential skill for students and professionals alike. In this article, we will explore the process of graphing the polynomial function $f(x)=x(3-x)(5-x)$ using parts (a) through (e). We will also determine the zero(s) and their multiplicity, and use this information to determine whether the graph crosses or touches the $x$-axis at each zero.

Part (a): Factor the Polynomial

The first step in graphing the polynomial function is to factor it. We can start by factoring out the greatest common factor (GCF) of the three terms:

f(x)=x(3−x)(5−x)f(x)=x(3-x)(5-x)

We can see that the GCF of the three terms is $x$, so we can factor it out:

f(x)=x(3−x)(5−x)=x(3−x)(5−x)f(x)=x(3-x)(5-x)=x(3-x)(5-x)

This tells us that the polynomial has a zero at $x=0$, since the term $x$ is equal to zero when $x=0$.

Part (b): Find the Zeros of the Polynomial

The next step is to find the zeros of the polynomial. We can do this by setting each factor equal to zero and solving for $x$:

x(3−x)(5−x)=0x(3-x)(5-x)=0

We can see that the first factor is equal to zero when $x=0$, which we already found in part (a). The second factor is equal to zero when $3-x=0$, which gives us $x=3$. The third factor is equal to zero when $5-x=0$, which gives us $x=5$.

Therefore, the zeros of the polynomial are $x=0$, $x=3$, and $x=5$.

Part (c): Determine the Multiplicity of Each Zero

The multiplicity of a zero is the number of times that the factor corresponding to that zero appears in the polynomial. In this case, the factor corresponding to the zero $x=0$ is $x$, which appears only once in the polynomial. Therefore, the multiplicity of the zero $x=0$ is 1.

The factor corresponding to the zero $x=3$ is $3-x$, which appears only once in the polynomial. Therefore, the multiplicity of the zero $x=3$ is 1.

The factor corresponding to the zero $x=5$ is $5-x$, which appears only once in the polynomial. Therefore, the multiplicity of the zero $x=5$ is 1.

Part (d): Determine Whether the Graph Crosses or Touches the $x$-Axis at Each Zero

The final step is to determine whether the graph crosses or touches the $x$-axis at each zero. We can do this by looking at the multiplicity of each zero.

If the multiplicity of a zero is 1, then the graph crosses the $x$-axis at that zero. If the multiplicity of a zero is greater than 1, then the graph touches the $x$-axis at that zero.

In this case, the multiplicity of each zero is 1, so the graph crosses the $x$-axis at each zero.

Conclusion

In conclusion, we have graphed the polynomial function $f(x)=x(3-x)(5-x)$ using parts (a) through (e). We have determined the zero(s) and their multiplicity, and used this information to determine whether the graph crosses or touches the $x$-axis at each zero.

The zeros of the polynomial are $x=0$, $x=3$, and $x=5$, and the multiplicity of each zero is 1. Therefore, the graph crosses the $x$-axis at each zero.

Graph of the Polynomial Function

Here is a graph of the polynomial function $f(x)=x(3-x)(5-x)$:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-1, 6, 400)
y = x*(3-x)*(5-x)


plt.plot(x, y)
plt.title('Graph of the Polynomial Function')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Introduction

Graphing polynomial functions is an essential skill for students and professionals alike. In our previous article, we explored the process of graphing the polynomial function $f(x)=x(3-x)(5-x)$ using parts (a) through (e). In this article, we will answer some frequently asked questions about graphing polynomial functions.

Q: What is a polynomial function?

A polynomial function is a function that can be written in the form $f(x)=a_nxn+a_{n-1}x{n-1}+\cdots+a_1x+a_0$, where $a_n\neq 0$ and $n$ is a non-negative integer.

Q: How do I graph a polynomial function?

To graph a polynomial function, you can follow these steps:

  1. Factor the polynomial function, if possible.
  2. Find the zeros of the polynomial function by setting each factor equal to zero and solving for $x$.
  3. Determine the multiplicity of each zero by counting the number of times that the factor corresponding to that zero appears in the polynomial.
  4. Determine whether the graph crosses or touches the $x$-axis at each zero by looking at the multiplicity of each zero.
  5. Plot the graph of the polynomial function using a graphing calculator or software.

Q: What is the difference between a zero and a root of a polynomial function?

A zero of a polynomial function is a value of $x$ that makes the polynomial function equal to zero. A root of a polynomial function is a value of $x$ that makes the polynomial function equal to zero, and is also a solution to the equation $f(x)=0$.

Q: How do I determine the multiplicity of a zero?

To determine the multiplicity of a zero, you can count the number of times that the factor corresponding to that zero appears in the polynomial. If the factor appears only once, then the multiplicity of the zero is 1. If the factor appears more than once, then the multiplicity of the zero is greater than 1.

Q: What is the difference between a graph that crosses the $x$-axis and a graph that touches the $x$-axis?

A graph that crosses the $x$-axis at a zero has a multiplicity of 1 at that zero. A graph that touches the $x$-axis at a zero has a multiplicity greater than 1 at that zero.

Q: How do I graph a polynomial function with a multiplicity greater than 1?

To graph a polynomial function with a multiplicity greater than 1, you can follow these steps:

  1. Factor the polynomial function, if possible.
  2. Find the zeros of the polynomial function by setting each factor equal to zero and solving for $x$.
  3. Determine the multiplicity of each zero by counting the number of times that the factor corresponding to that zero appears in the polynomial.
  4. Plot the graph of the polynomial function using a graphing calculator or software.

Q: What are some common mistakes to avoid when graphing polynomial functions?

Some common mistakes to avoid when graphing polynomial functions include:

  • Not factoring the polynomial function, if possible.
  • Not finding the zeros of the polynomial function.
  • Not determining the multiplicity of each zero.
  • Not plotting the graph of the polynomial function using a graphing calculator or software.

Conclusion

Graphing polynomial functions is an essential skill for students and professionals alike. By following the steps outlined in this article, you can graph polynomial functions with ease. Remember to factor the polynomial function, find the zeros of the polynomial function, determine the multiplicity of each zero, and plot the graph of the polynomial function using a graphing calculator or software.

Common Polynomial Functions

Here are some common polynomial functions:

  • f(x)=x2+1f(x)=x^2+1

  • f(x)=x3−2x2+x+1f(x)=x^3-2x^2+x+1

  • f(x)=x4−3x2+2f(x)=x^4-3x^2+2

Graphing Polynomial Functions with Technology

Here is an example of how to graph a polynomial function using a graphing calculator or software:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-10, 10, 400)
y = x**2 + 1


plt.plot(x, y)
plt.title('Graph of the Polynomial Function')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

This code will graph the polynomial function $f(x)=x^2+1$ using a graphing calculator or software.