Simplify: $ \frac{x^3 - 18x^2 + 81x}{x^3 - 9x^2} $

Mar 10, 2025 by ADMIN 51 views

$Simplify: $\frac{x^3 - 18x^2 + 81x}{x^3 - 9x^2}$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved in simplifying rational expressions. In this article, we will focus on simplifying the given rational expression $\frac{x^3 - 18x^2 + 81x}{x^3 - 9x^2}$ . We will use various techniques such as factoring, cancelling common factors, and simplifying the resulting expression.

Step 1: Factor the Numerator and Denominator

To simplify the given rational expression, we first need to factor the numerator and denominator. The numerator can be factored as follows:

x^3 - 18x^2 + 81x = x(x^2 - 18x + 81)

We can further factor the quadratic expression inside the parentheses as:

x^2 - 18x + 81 = (x - 9)^2

So, the numerator can be written as:

x(x - 9)^2

The denominator can be factored as follows:

x^3 - 9x^2 = x^2(x - 9)

Step 2: Cancel Common Factors

Now that we have factored the numerator and denominator, we can cancel common factors. We can see that both the numerator and denominator have a common factor of $x(x - 9)$ . We can cancel this common factor as follows:

\frac{x(x - 9)^2}{x^2(x - 9)} = \frac{(x - 9)}{x}

Step 3: Simplify the Resulting Expression

The resulting expression $\frac{(x - 9)}{x}$ is already simplified, but we can further simplify it by noticing that the numerator and denominator have a common factor of $(x - 9)$ . However, we cannot cancel this common factor because it would result in a fraction with a denominator of zero, which is undefined.

Conclusion

In conclusion, we have simplified the given rational expression $\frac{x^3 - 18x^2 + 81x}{x^3 - 9x^2}$ by factoring the numerator and denominator, cancelling common factors, and simplifying the resulting expression. The simplified expression is $\frac{(x - 9)}{x}$ .

Final Answer

The final answer is $\boxed{\frac{x-9}{x}}$ .

Applications of Simplifying Rational Expressions

Simplifying rational expressions has numerous applications in mathematics and other fields. Some of the applications include:

Algebra: Simplifying rational expressions is a crucial skill in algebra, and it's used to solve equations and inequalities.
Calculus: Simplifying rational expressions is used in calculus to find derivatives and integrals.
Engineering: Simplifying rational expressions is used in engineering to design and analyze systems.
Computer Science: Simplifying rational expressions is used in computer science to optimize algorithms and data structures.

Tips and Tricks for Simplifying Rational Expressions

Here are some tips and tricks for simplifying rational expressions:

Factor the numerator and denominator: Factoring the numerator and denominator can help you cancel common factors and simplify the expression.
Cancel common factors: Cancelling common factors can help you simplify the expression and make it easier to work with.
Simplify the resulting expression: Simplifying the resulting expression can help you make it easier to work with and understand.
Use algebraic identities: Using algebraic identities such as the difference of squares can help you simplify the expression and make it easier to work with.

Common Mistakes to Avoid When Simplifying Rational Expressions

Here are some common mistakes to avoid when simplifying rational expressions:

Not factoring the numerator and denominator: Not factoring the numerator and denominator can make it difficult to cancel common factors and simplify the expression.
Not cancelling common factors: Not cancelling common factors can make the expression more complicated and difficult to work with.
Not simplifying the resulting expression: Not simplifying the resulting expression can make it difficult to understand and work with.
Using incorrect algebraic identities: Using incorrect algebraic identities can lead to incorrect simplifications and make the expression more complicated.

Real-World Examples of Simplifying Rational Expressions

Here are some real-world examples of simplifying rational expressions:

Optimizing algorithms: Simplifying rational expressions can help optimize algorithms and make them more efficient.
Designing systems: Simplifying rational expressions can help design systems and make them more efficient.
Analyzing data: Simplifying rational expressions can help analyze data and make it easier to understand.
Solving equations: Simplifying rational expressions can help solve equations and make it easier to understand the solution.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill in mathematics and other fields. It's used to solve equations and inequalities, find derivatives and integrals, design and analyze systems, and optimize algorithms and data structures. By following the tips and tricks outlined in this article, you can simplify rational expressions and make them easier to work with.

Introduction

In our previous article, we discussed how to simplify the rational expression $\frac{x^3 - 18x^2 + 81x}{x^3 - 9x^2}$ . We used various techniques such as factoring, cancelling common factors, and simplifying the resulting expression. In this article, we will answer some frequently asked questions about simplifying rational expressions.

Q&A

Q: What is the first step in simplifying a rational expression?

A: The first step in simplifying a rational expression is to factor the numerator and denominator.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, you can use various techniques such as factoring out common factors, using algebraic identities, and factoring quadratic expressions.

Q: What is the difference between factoring and cancelling common factors?

A: Factoring involves breaking down an expression into its simplest form, while cancelling common factors involves removing common factors from the numerator and denominator.

Q: Can I cancel common factors if the numerator and denominator have a common factor of zero?

A: No, you cannot cancel common factors if the numerator and denominator have a common factor of zero. This is because dividing by zero is undefined.

Q: How do I simplify the resulting expression after cancelling common factors?

A: After cancelling common factors, you can simplify the resulting expression by combining like terms and using algebraic identities.

Q: What are some common mistakes to avoid when simplifying rational expressions?

A: Some common mistakes to avoid when simplifying rational expressions include not factoring the numerator and denominator, not cancelling common factors, and not simplifying the resulting expression.

Q: How do I know if I have simplified a rational expression correctly?

A: To check if you have simplified a rational expression correctly, you can plug in values for the variable and check if the expression is true.

Q: Can I use technology to simplify rational expressions?

A: Yes, you can use technology such as calculators and computer software to simplify rational expressions.

Q: How do I apply simplifying rational expressions in real-world scenarios?

A: Simplifying rational expressions is used in various real-world scenarios such as optimizing algorithms, designing systems, analyzing data, and solving equations.