Writing the Function in Completely Factored Form

Oct 13, 2025 by wbrum.org 17 views

=====================================================

Introduction

In this article, we will delve into the world of rational functions and explore the process of factoring and analyzing a given function. We will examine the function $ f(x)=\frac{\left(x^{2-9x+18\right)(x+2)}{\left(x}2-2x+1\right)(x-3)} $ and answer four key questions: (1) write the function in completely factored form, (2) determine the coordinates of zero(s), (3) determine the coordinates of the y-intercept, and (4) discuss the significance of these findings in the context of mathematics.

Factoring the Function

To factor the function $ f(x)=\frac{\left(x^{2-9x+18\right)(x+2)}{\left(x}2-2x+1\right)(x-3)} $, we need to start by factoring the numerator and denominator separately.

Factoring the Numerator

The numerator of the function is given by $ x^2-9x+18 $. We can factor this quadratic expression by finding two numbers whose product is $ 18 $ and whose sum is $ -9 $. These numbers are $ -6 $ and $ -3 $, so we can write the numerator as:

x^2-9x+18 = (x-6)(x-3)

Next, we need to factor the expression $ (x+2) $, which is already in factored form.

Factoring the Denominator

The denominator of the function is given by $ x^2-2x+1 $. We can factor this quadratic expression by finding two numbers whose product is $ 1 $ and whose sum is $ -2 $. These numbers are $ -1 $ and $ -1 $, so we can write the denominator as:

x^2-2x+1 = (x-1)^2

Finally, we need to factor the expression $ (x-3) $, which is already in factored form.

Writing the Function in Completely Factored Form

Now that we have factored the numerator and denominator, we can write the function in completely factored form:

f(x) = \frac{(x-6)(x-3)(x+2)}{(x-1)^2(x-3)}

We can simplify this expression by canceling out the common factor of $ (x-3) $:

f(x) = \frac{(x-6)(x+2)}{(x-1)^2}

Determining the Coordinates of Zero(s)

To determine the coordinates of zero(s), we need to set the numerator of the function equal to zero and solve for $ x $. This gives us the equation:

(x-6)(x+2) = 0

Solving for $ x $, we get:

x = 6 \text{ or } x = -2

Therefore, the coordinates of zero(s) are $ (6, 0) $ and $ (-2, 0) $.

Determining the Coordinates of the Y-Intercept

To determine the coordinates of the y-intercept, we need to set $ x = 0 $ and solve for $ y $. This gives us the equation:

f(0) = \frac{(0-6)(0+2)}{(0-1)^2}

Simplifying this expression, we get:

f(0) = \frac{(-6)(2)}{(-1)^2}

f(0) = \frac{-12}{1}

f(0) = -12

Therefore, the coordinates of the y-intercept are $ (0, -12) $.

Discussion

In this article, we have explored the process of factoring and analyzing a rational function. We have written the function in completely factored form, determined the coordinates of zero(s), and determined the coordinates of the y-intercept. These findings have significant implications in the context of mathematics, particularly in the fields of algebra and calculus.

The process of factoring a rational function involves breaking down the numerator and denominator into their simplest forms. This can be a complex and time-consuming process, but it is essential for understanding the behavior of the function.

The coordinates of zero(s) are critical in understanding the behavior of a function. They represent the points where the function intersects the x-axis, and they can be used to determine the intervals where the function is positive or negative.

The coordinates of the y-intercept are also essential in understanding the behavior of a function. They represent the point where the function intersects the y-axis, and they can be used to determine the value of the function at a given point.

In conclusion, the process of factoring and analyzing a rational function is a complex and nuanced process that requires a deep understanding of algebra and calculus. However, the rewards are well worth the effort, as it can provide valuable insights into the behavior of the function and its applications in mathematics and other fields.

Conclusion

=====================================================

Introduction

In our previous article, we explored the process of factoring and analyzing a rational function. We wrote the function in completely factored form, determined the coordinates of zero(s), and determined the coordinates of the y-intercept. In this article, we will answer some frequently asked questions (FAQs) related to factoring and analyzing a rational function.

Q: What is a rational function?

A rational function is a function that can be expressed as the ratio of two polynomials. It is typically written in the form:

f(x) = \frac{p(x)}{q(x)}

where $ p(x) $ and $ q(x) $ are polynomials.

Q: How do I factor a rational function?

To factor a rational function, you need to factor the numerator and denominator separately. This involves breaking down the polynomials into their simplest forms and canceling out any common factors.

Q: What are the coordinates of zero(s)?

The coordinates of zero(s) are the points where the function intersects the x-axis. To find the coordinates of zero(s), you need to set the numerator of the function equal to zero and solve for $ x $.

Q: What are the coordinates of the y-intercept?

The coordinates of the y-intercept are the point where the function intersects the y-axis. To find the coordinates of the y-intercept, you need to set $ x = 0 $ and solve for $ y $.

Q: How do I determine the intervals where the function is positive or negative?

To determine the intervals where the function is positive or negative, you need to examine the signs of the factors in the numerator and denominator. If the signs of the factors are the same, the function is positive. If the signs of the factors are different, the function is negative.

Q: What are the applications of factoring and analyzing a rational function?

Factoring and analyzing a rational function has numerous applications in mathematics and other fields. Some of the applications include:

Algebra: Factoring and analyzing a rational function is essential in algebra, particularly in the study of polynomial equations and functions.
Calculus: Factoring and analyzing a rational function is also essential in calculus, particularly in the study of limits, derivatives, and integrals.
Engineering: Factoring and analyzing a rational function is used in engineering to model and analyze complex systems, such as electrical circuits and mechanical systems.
Economics: Factoring and analyzing a rational function is used in economics to model and analyze economic systems, such as supply and demand curves.

Q: What are some common mistakes to avoid when factoring and analyzing a rational function?

Some common mistakes to avoid when factoring and analyzing a rational function include:

Not factoring the numerator and denominator separately: This can lead to incorrect results and a lack of understanding of the function's behavior.
Not canceling out common factors: This can lead to incorrect results and a lack of understanding of the function's behavior.
Not examining the signs of the factors: This can lead to incorrect results and a lack of understanding of the function's behavior.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to factoring and analyzing a rational function. We hope that this article has provided a comprehensive and informative overview of the process of factoring and analyzing a rational function. If you have any further questions or need additional clarification, please don't hesitate to ask.

Additional Resources

For additional resources and information on factoring and analyzing a rational function, please see the following:

Textbooks: There are many excellent textbooks on algebra and calculus that cover the topic of factoring and analyzing a rational function.
Online Resources: There are many online resources, such as Khan Academy and MIT OpenCourseWare, that provide video lectures and tutorials on factoring and analyzing a rational function.
Software: There are many software programs, such as Mathematica and Maple, that can be used to factor and analyze a rational function.

We hope that this article has been helpful in your understanding of factoring and analyzing a rational function. If you have any further questions or need additional clarification, please don't hesitate to ask.