Properties of Rational Roots

Oct 13, 2025 by windailys.com 138 views

Introduction

When dealing with polynomial functions, understanding the behavior of their graphs is crucial in identifying the roots of the function. In this case, we are given the function $f(x) = 4x^3 - 13x^2 + 9x + 2$ and its graph. Our task is to determine how many roots of $f(x)$ are rational numbers. To approach this problem, we need to understand the properties of rational roots and how they relate to the graph of the function.

Properties of Rational Roots

A rational root of a polynomial function is a root that can be expressed as a fraction of two integers, i.e., $p/q$ , where $p$ and $q$ are integers and $q$ is non-zero. The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial function $f(x)$ , then $p$ must be a factor of the constant term of $f(x)$ , and $q$ must be a factor of the leading coefficient of $f(x)$ .

Factors of the Constant Term and Leading Coefficient

To apply the Rational Root Theorem, we need to find the factors of the constant term and the leading coefficient of the function $f(x) = 4x^3 - 13x^2 + 9x + 2$ . The constant term is 2, and its factors are $\pm 1$ and $\pm 2$ . The leading coefficient is 4, and its factors are $\pm 1$ , $\pm 2$ , and $\pm 4$ .

Possible Rational Roots

Using the factors of the constant term and the leading coefficient, we can list the possible rational roots of the function $f(x)$ . These are:

$\pm 1$
$\pm 2$
$\pm 1/2$
$\pm 1/4$

Graph Analysis

Now that we have identified the possible rational roots, we need to analyze the graph of the function $f(x)$ to determine which of these roots are actual roots of the function. By examining the graph, we can see that there are three x-intercepts, which correspond to the roots of the function.

Identifying Rational Roots

To determine which of the possible rational roots are actual roots of the function, we need to substitute each of the possible roots into the function $f(x)$ and check if the result is zero. We can start by substituting the possible rational roots into the function and checking if the result is zero.

Substitution and Evaluation

Let's substitute each of the possible rational roots into the function $f(x)$ and evaluate the result.

For $x = 1$ : $f(1) = 4(1)^3 - 13(1)^2 + 9(1) + 2 = 4 - 13 + 9 + 2 = 2 \neq 0$
For $x = -1$ : $f(-1) = 4(-1)^3 - 13(-1)^2 + 9(-1) + 2 = -4 - 13 - 9 + 2 = -24 \neq 0$
For $x = 2$ : $f(2) = 4(2)^3 - 13(2)^2 + 9(2) + 2 = 32 - 52 + 18 + 2 = 0$
For $x = -2$ : $f(-2) = 4(-2)^3 - 13(-2)^2 + 9(-2) + 2 = -32 - 52 - 18 + 2 = -100 \neq 0$
For $x = 1/2$ : $f(1/2) = 4(1/2)^3 - 13(1/2)^2 + 9(1/2) + 2 = 1/4 - 13/4 + 9/2 + 2 = 0$
For $x = -1/2$ : $f(-1/2) = 4(-1/2)^3 - 13(-1/2)^2 + 9(-1/2) + 2 = -1/4 - 13/4 - 9/2 + 2 = -25/2 \neq 0$
For $x = 1/4$ : $f(1/4) = 4(1/4)^3 - 13(1/4)^2 + 9(1/4) + 2 = 1/64 - 13/16 + 9/4 + 2 = 0$
For $x = -1/4$ : $f(-1/4) = 4(-1/4)^3 - 13(-1/4)^2 + 9(-1/4) + 2 = -1/64 - 13/16 - 9/4 + 2 = -25/16 \neq 0$

Conclusion

Based on the graph analysis and the substitution and evaluation of the possible rational roots, we can conclude that the function $f(x) = 4x^3 - 13x^2 + 9x + 2$ has three rational roots: $x = 2$ , $x = 1/2$ , and $x = 1/4$ . Therefore, the number of rational roots of the function $f(x)$ is 3.

Final Thoughts

In conclusion, the graph of the function $f(x) = 4x^3 - 13x^2 + 9x + 2$ provides valuable information about the behavior of the function and its roots. By analyzing the graph and applying the Rational Root Theorem, we can determine the number of rational roots of the function. In this case, we found that the function has three rational roots: $x = 2$ , $x = 1/2$ , and $x = 1/4$ . This demonstrates the importance of graph analysis and the Rational Root Theorem in understanding the properties of polynomial functions.

Introduction

In our previous article, we analyzed the graph of the function $f(x) = 4x^3 - 13x^2 + 9x + 2$ and determined that it has three rational roots: $x = 2$ , $x = 1/2$ , and $x = 1/4$ . In this article, we will answer some frequently asked questions about the graph of the function and its roots.

Q: What is the significance of the Rational Root Theorem in understanding the graph of the function?

A: The Rational Root Theorem is a crucial tool in understanding the graph of a polynomial function. It states that if a rational number $p/q$ is a root of the polynomial function $f(x)$ , then $p$ must be a factor of the constant term of $f(x)$ , and $q$ must be a factor of the leading coefficient of $f(x)$ . This theorem helps us identify the possible rational roots of the function and narrow down the search for actual roots.

Q: How do you determine the number of rational roots of a polynomial function?

A: To determine the number of rational roots of a polynomial function, we need to analyze the graph of the function and apply the Rational Root Theorem. We can start by finding the factors of the constant term and the leading coefficient of the function. Then, we can list the possible rational roots and substitute each of them into the function to check if the result is zero. By doing this, we can identify the actual rational roots of the function.

Q: What is the relationship between the graph of the function and its roots?

A: The graph of a polynomial function provides valuable information about the behavior of the function and its roots. By analyzing the graph, we can identify the x-intercepts, which correspond to the roots of the function. We can also use the graph to determine the number of rational roots of the function.

Q: How do you use the graph of the function to identify the rational roots?

A: To use the graph of the function to identify the rational roots, we need to analyze the x-intercepts of the graph. The x-intercepts correspond to the roots of the function, and we can use the Rational Root Theorem to determine which of these roots are rational.

Q: What are some common mistakes to avoid when analyzing the graph of a polynomial function?

A: Some common mistakes to avoid when analyzing the graph of a polynomial function include:

Not considering the Rational Root Theorem when identifying possible rational roots
Not analyzing the graph of the function carefully to identify the x-intercepts
Not substituting each possible rational root into the function to check if the result is zero
Not considering the possibility of complex roots

Q: How do you determine the number of complex roots of a polynomial function?

A: To determine the number of complex roots of a polynomial function, we need to analyze the graph of the function and apply the Complex Conjugate Root Theorem. This theorem states that if a complex number $a + bi$ is a root of the polynomial function $f(x)$ , then its complex conjugate $a - bi$ is also a root of the function.

Q: What is the relationship between the graph of the function and its complex roots?

A: The graph of a polynomial function provides valuable information about the behavior of the function and its complex roots. By analyzing the graph, we can identify the complex roots of the function and determine their multiplicity.

Q: How do you use the graph of the function to identify the complex roots?

A: To use the graph of the function to identify the complex roots, we need to analyze the graph carefully to identify the complex x-intercepts. The complex x-intercepts correspond to the complex roots of the function, and we can use the Complex Conjugate Root Theorem to determine which of these roots are complex.

Q: What are some common applications of the graph of a polynomial function?

A: Some common applications of the graph of a polynomial function include:

Modeling real-world phenomena, such as population growth or chemical reactions
Determining the maximum or minimum value of a function
Identifying the x-intercepts of a function, which correspond to the roots of the function
Analyzing the behavior of a function over a given interval

Q: How do you use the graph of the function to model real-world phenomena?

A: To use the graph of the function to model real-world phenomena, we need to analyze the graph carefully to identify the x-intercepts, which correspond to the roots of the function. We can then use the graph to determine the maximum or minimum value of the function and analyze its behavior over a given interval.

Q: What are some common challenges when analyzing the graph of a polynomial function?

A: Some common challenges when analyzing the graph of a polynomial function include:

Difficulty in identifying the x-intercepts of the graph
Difficulty in determining the number of rational or complex roots of the function
Difficulty in analyzing the behavior of the function over a given interval

Q: How do you overcome these challenges when analyzing the graph of a polynomial function?

A: To overcome these challenges, we need to analyze the graph carefully and apply the Rational Root Theorem and the Complex Conjugate Root Theorem. We can also use technology, such as graphing calculators or computer software, to help us analyze the graph and determine the number of rational or complex roots of the function.

Conclusion

In conclusion, the graph of a polynomial function provides valuable information about the behavior of the function and its roots. By analyzing the graph and applying the Rational Root Theorem and the Complex Conjugate Root Theorem, we can determine the number of rational and complex roots of the function. We can also use the graph to model real-world phenomena and analyze the behavior of the function over a given interval.