Simplify The Expression:$ \frac{9x^2 - X^3}{x^2 - 3x - 54} $

$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it plays a vital role in solving various mathematical problems. In this article, we will focus on simplifying the given expression: $\frac{9x^2 - x^3}{x^2 - 3x - 54}$. We will use various algebraic techniques to simplify the expression and make it more manageable.

Step 1: Factor the Numerator and Denominator

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

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

The denominator can be factored as follows:

$$x^2 - 3x - 54 = (x - 9)(x + 6)$$

Step 2: Cancel Common Factors

Now that we have factored the numerator and denominator, we can cancel common factors. The expression can be rewritten as:

$$\frac{x^2(9 - x)}{(x - 9)(x + 6)}$$

We can cancel the common factor $(x - 9)$ from the numerator and denominator:

$$\frac{x^2(9 - x)}{(x - 9)(x + 6)} = \frac{x^2(9 - x)}{x + 6}$$

Step 3: Simplify the Expression

Now that we have canceled the common factor, we can simplify the expression further. We can rewrite the expression as:

$$\frac{x^2(9 - x)}{x + 6} = \frac{9x^2 - x^3}{x + 6}$$

Step 4: Factor the Numerator

We can factor the numerator as follows:

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

Step 5: Cancel Common Factors

We can cancel the common factor $x^2$ from the numerator and denominator:

$$\frac{x^2(9 - x)}{x + 6} = \frac{9 - x}{x + 6}$$

Step 6: Simplify the Expression

Now that we have canceled the common factor, we can simplify the expression further. We can rewrite the expression as:

$$\frac{9 - x}{x + 6}$$

Conclusion

In this article, we have simplified the given expression: $\frac{9x^2 - x^3}{x^2 - 3x - 54}$. We used various algebraic techniques to simplify the expression and make it more manageable. We factored the numerator and denominator, canceled common factors, and simplified the expression further. The final simplified expression is $\frac{9 - x}{x + 6}$.

Final Answer

The final answer is $\boxed{\frac{9 - x}{x + 6}}$.

Related Topics

• Factoring algebraic expressions
• Canceling common factors
• Simplifying algebraic expressions

References

• [1] Algebraic Expressions, Khan Academy
• [2] Factoring and Canceling, Mathway
• [3] Simplifying Algebraic Expressions, Purplemath

Keywords

• Simplifying algebraic expressions
• Factoring algebraic expressions
• Canceling common factors
• Algebraic techniques
• Mathematical problems
• Algebraic expressions
• Simplification
• Factoring
• Canceling
• Algebra
  

$$

Introduction

In our previous article, we simplified the given expression: $\frac{9x^2 - x^3}{x^2 - 3x - 54}$. We used various algebraic techniques to simplify the expression and make it more manageable. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q1: What is the first step in simplifying the expression?

A1: The first step in simplifying the expression is to factor the numerator and denominator. We can factor the numerator as follows: $9x^2 - x^3 = x^2(9 - x)$, and the denominator as follows: $x^2 - 3x - 54 = (x - 9)(x + 6)$.

Q2: How do we cancel common factors in the expression?

A2: We can cancel common factors in the expression by identifying the common factors in the numerator and denominator. In this case, we can cancel the common factor $(x - 9)$ from the numerator and denominator.

Q3: What is the final simplified expression?

A3: The final simplified expression is $\frac{9 - x}{x + 6}$.

Q4: How do we factor the numerator?

A4: We can factor the numerator as follows: $9x^2 - x^3 = x^2(9 - x)$.

Q5: What is the importance of simplifying algebraic expressions?

A5: Simplifying algebraic expressions is crucial in solving various mathematical problems. It helps us to make the expression more manageable and easier to work with.

Q6: Can we simplify the expression further?

A6: Yes, we can simplify the expression further by canceling common factors and simplifying the expression.

Q7: What are some common algebraic techniques used to simplify expressions?

A7: Some common algebraic techniques used to simplify expressions include factoring, canceling common factors, and simplifying the expression.

Q8: How do we identify common factors in the numerator and denominator?

A8: We can identify common factors in the numerator and denominator by looking for common terms or factors.

Q9: What is the final answer to the expression?

A9: The final answer to the expression is $\boxed{\frac{9 - x}{x + 6}}$.

Q10: Can we use the simplified expression to solve mathematical problems?

A10: Yes, we can use the simplified expression to solve mathematical problems. The simplified expression can be used as a starting point to solve various mathematical problems.

Conclusion

In this article, we have answered some frequently asked questions related to the simplification of the expression: $\frac{9x^2 - x^3}{x^2 - 3x - 54}$. We have discussed various algebraic techniques used to simplify the expression and provided answers to some common questions.

Final Answer

The final answer is $\boxed{\frac{9 - x}{x + 6}}$.

Related Topics

• Factoring algebraic expressions
• Canceling common factors
• Simplifying algebraic expressions
• Algebraic techniques
• Mathematical problems
• Algebraic expressions
• Simplification
• Factoring
• Canceling
• Algebra

References

• [1] Algebraic Expressions, Khan Academy
• [2] Factoring and Canceling, Mathway
• [3] Simplifying Algebraic Expressions, Purplemath

Keywords

• Simplifying algebraic expressions
• Factoring algebraic expressions
• Canceling common factors
• Algebraic techniques
• Mathematical problems
• Algebraic expressions
• Simplification
• Factoring
• Canceling
• Algebra