Simplify The Expression Completely If Possible. (x³ + 2x²) / (x² - 12x + 27)

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Introduction

In this article, we will simplify the given expression completely if possible. The expression is a rational expression, which is a fraction of two polynomials. We will use various techniques to simplify the expression, including factoring and canceling common factors.

The Given Expression

The given expression is:

(x³ + 2x²) / (x² - 12x + 27)

Step 1: Factor the Numerator

To simplify the expression, we first need to factor the numerator. The numerator is a quadratic expression, which can be factored as:

x² + 2x = x(x + 2)

However, we also have an x term in the numerator, so we can rewrite the numerator as:

x(x + 2) + 2x² = x(x + 2) + 2x(x)

Now, we can factor out the common term x:

x(x + 2 + 2x) = x(x + 2x + 2)

Combine like terms:

x(3x + 2) = x(3x + 2)

Step 2: Factor the Denominator

Next, we need to factor the denominator. The denominator is a quadratic expression, which can be factored as:

x² - 12x + 27 = (x - 3)(x - 9)

Step 3: Simplify the Expression

Now that we have factored the numerator and denominator, we can simplify the expression by canceling common factors. The numerator is x(3x + 2) and the denominator is (x - 3)(x - 9). We can cancel the common factor (x - 3) from the numerator and denominator:

(x(3x + 2)) / ((x - 3)(x - 9)) = (3x + 2) / (x - 9)

Step 4: Check for Common Factors

We have simplified the expression, but we need to check if there are any common factors that can be canceled. The numerator is 3x + 2 and the denominator is x - 9. There are no common factors, so the expression is simplified.

Conclusion

In this article, we simplified the given expression completely if possible. We factored the numerator and denominator, and then canceled common factors to simplify the expression. The final simplified expression is:

(3x + 2) / (x - 9)

Final Answer

The final answer is:

(3x + 2) / (x - 9)

Example Use Case

This expression can be used in various mathematical applications, such as solving equations or graphing functions. For example, if we want to find the value of x that makes the expression equal to 1, we can set up the equation:

(3x + 2) / (x - 9) = 1

We can then solve for x by multiplying both sides of the equation by (x - 9):

3x + 2 = x - 9

Subtracting x from both sides gives:

2x + 2 = -9

Subtracting 2 from both sides gives:

2x = -11

Dividing both sides by 2 gives:

x = -11/2Therefore, the value of x that makes the expression equal to 1 is x = -11/2.

Tips and Tricks

When simplifying rational expressions, it's essential to factor the numerator and denominator and then cancel common factors. This can help simplify the expression and make it easier to work with. Additionally, be sure to check for common factors after simplifying the expression to ensure that it is fully simplified.

Related Topics

  • Simplifying rational expressions
  • Factoring quadratic expressions
  • Canceling common factors
  • Solving equations
  • Graphing functions

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman

Note: The references provided are for general information purposes only and are not specific to this article.

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Introduction

In our previous article, we simplified the given expression completely if possible. We factored the numerator and denominator, and then canceled common factors to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying rational expressions.

Q&A

Q: What is a rational expression?

A: A rational expression is a fraction of two polynomials. It is a way of expressing a relationship between two variables using a fraction.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, and then cancel common factors. This can help simplify the expression and make it easier to work with.

Q: What is factoring?

A: Factoring is the process of expressing a polynomial as a product of simpler polynomials. For example, the polynomial x² + 4x + 4 can be factored as (x + 2)(x + 2).

Q: What is canceling common factors?

A: Canceling common factors is the process of removing common factors from the numerator and denominator of a rational expression. For example, if you have the expression (x + 2) / (x + 2), you can cancel the common factor (x + 2) to get 1.

Q: Why is it important to simplify rational expressions?

A: Simplifying rational expressions is important because it can help make the expression easier to work with. It can also help you avoid mistakes when solving equations or graphing functions.

Q: Can I simplify a rational expression if it has no common factors?

A: Yes, you can still simplify a rational expression even if it has no common factors. You can try to factor the numerator and denominator, or use other techniques such as multiplying by a conjugate.

Q: How do I know if a rational expression is fully simplified?

A: To know if a rational expression is fully simplified, you need to check if there are any common factors that can be canceled. You can also try to factor the numerator and denominator, or use other techniques such as multiplying by a conjugate.

Q: Can I use a calculator to simplify a rational expression?

A: Yes, you can use a calculator to simplify a rational expression. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Example Questions

Q: Simplify the expression (x² + 4x + 4) / (x² - 4x + 4)

A: To simplify this expression, we need to factor the numerator and denominator. The numerator can be factored as (x + 2)(x + 2), and the denominator can be factored as (x - 2)(x - 2). We can then cancel the common factor (x - 2) to get (x + 2) / (x - 2).

Q: Simplify the expression (x³ + 2x²) / (x² - 12x + 27)

A: To simplify this expression, we need to factor the numerator and denominator. The numerator can be factored as x(x + 2), and the denominator can be factored as (x - 3)(x - 9). We can then cancel the common factor (x - 3) to get (3x + 2) / (x - 9).

Tips and Tricks

  • Always check your work by hand to make sure you get the correct answer.
  • Use a calculator to simplify rational expressions, but also check your work by hand.
  • Factor the numerator and denominator to simplify the expression.
  • Cancel common factors to simplify the expression.
  • Use other techniques such as multiplying by a conjugate to simplify the expression.

Related Topics

  • Simplifying rational expressions
  • Factoring quadratic expressions
  • Canceling common factors
  • Solving equations
  • Graphing functions

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman

Note: The references provided are for general information purposes only and are not specific to this article.