Simplify: x⁴ / (x - 3) - 81 / (x - 3) A) (x + 3)(x² - 9) B) (x - 3)(x² + 9) C) (x + 3)(x² + 9) D) (x - 3)(x² - 9)

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Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in calculus and algebra. It involves breaking down complex expressions into simpler ones, making it easier to solve equations and manipulate variables. In this article, we will focus on simplifying a specific expression involving polynomials and fractions.

The Expression to Simplify

The given expression is:

x⁴ / (x - 3) - 81 / (x - 3)

Our goal is to simplify this expression by combining the two fractions and factoring out common terms.

Step 1: Combine the Fractions

To combine the fractions, we need to find a common denominator. In this case, the common denominator is (x - 3). We can rewrite the expression as:

(x⁴ - 81) / (x - 3)

Step 2: Factor the Numerator

The numerator, x⁴ - 81, can be factored using the difference of squares formula:

a² - b² = (a + b)(a - b)

In this case, a = x² and b = 9. So, we can factor the numerator as:

(x² + 9)(x² - 9)

Step 3: Factor the Denominator

The denominator, x - 3, is already factored.

Step 4: Simplify the Expression

Now that we have factored the numerator and denominator, we can simplify the expression by canceling out common factors. In this case, we can cancel out the (x² - 9) term:

(x² + 9)(x² - 9) / (x - 3) = (x² + 9)

However, we are not done yet. We need to consider the original expression, which had a negative sign in front of the second term. To account for this, we can multiply the entire expression by -1:

-(x² + 9) = -(x² + 9)

Step 5: Factor the Expression

Now that we have simplified the expression, we can factor it further by recognizing that it is a difference of squares:

-(x² + 9) = -(x² + 3²)

Using the difference of squares formula, we can factor the expression as:

-(x + 3)(x - 3)

However, we are not done yet. We need to consider the original expression, which had a negative sign in front of the second term. To account for this, we can multiply the entire expression by -1:

(x + 3)(x - 3)

Conclusion

In conclusion, the simplified expression is:

(x + 3)(x - 3)

This is the correct answer.

Answer Key

The correct answer is:

D) (x - 3)(x² - 9)

However, this is not the correct answer. The correct answer is:

C) (x + 3)(x² + 9)

Discussion

This problem requires a deep understanding of algebraic expressions and factoring. It also requires the ability to recognize and apply the difference of squares formula. The correct answer is not immediately obvious, and it requires careful analysis and manipulation of the expression.

**Tips and Tricks-------------------

  • When simplifying complex algebraic expressions, it is essential to break down the expression into smaller parts and focus on one part at a time.
  • Use the difference of squares formula to factor expressions of the form a² - b².
  • Be careful when multiplying and dividing fractions, and make sure to cancel out common factors.
  • Use algebraic manipulations to simplify expressions and make them easier to work with.

Practice Problems

  • Simplify the expression: x³ / (x + 2) - 27 / (x + 2)
  • Simplify the expression: x² / (x - 4) + 16 / (x - 4)
  • Simplify the expression: x⁴ / (x - 1) - 100 / (x - 1)

References

Introduction

In our previous article, we explored the process of simplifying complex algebraic expressions. We walked through a step-by-step example of how to simplify the expression x⁴ / (x - 3) - 81 / (x - 3). In this article, we will answer some frequently asked questions about simplifying complex algebraic expressions.

Q: What is the difference between simplifying and factoring?

A: Simplifying and factoring are two related but distinct concepts in algebra. Factoring involves breaking down an expression into its constituent parts, while simplifying involves reducing an expression to its simplest form. For example, the expression x² + 5x + 6 can be factored as (x + 3)(x + 2), but it can also be simplified as (x + 3)(x + 2) = x² + 5x + 6.

Q: How do I know when to simplify an expression?

A: You should simplify an expression whenever possible. Simplifying expressions can make them easier to work with, and it can also help you to identify patterns and relationships between variables. In general, you should simplify expressions whenever you are working with them in a mathematical context.

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

A: Some common mistakes to avoid when simplifying expressions include:

  • Not canceling out common factors
  • Not using the correct order of operations
  • Not simplifying expressions that can be simplified
  • Not checking for errors in the simplification process

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, you should follow these steps:

  1. Find a common denominator for the fractions
  2. Add or subtract the numerators
  3. Simplify the resulting expression

For example, the expression x⁴ / (x - 3) - 81 / (x - 3) can be simplified as follows:

  1. Find a common denominator: (x - 3)
  2. Add the numerators: x⁴ - 81
  3. Simplify the resulting expression: (x⁴ - 81) / (x - 3)

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you should follow these steps:

  1. Simplify the exponents individually
  2. Combine the simplified exponents using the rules of exponentiation

For example, the expression x²y³ can be simplified as follows:

  1. Simplify the exponents individually: x² and y³
  2. Combine the simplified exponents: x²y³ = (xy)³

Q: What are some common algebraic identities that can be used to simplify expressions?

A: Some common algebraic identities that can be used to simplify expressions include:

  • a² - b² = (a + b)(a - b)
  • a² + b² = (a + b)² - 2ab
  • (a + b)² = a² + 2ab + b²
  • (a - b)² = a² - 2ab + b²

These identities can be to simplify expressions by factoring or expanding them.

Conclusion

In conclusion, simplifying complex algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article, you can simplify expressions and make them easier to work with. Remember to avoid common mistakes, such as not canceling out common factors or not using the correct order of operations. With practice and experience, you will become more comfortable simplifying expressions and will be able to tackle even the most complex algebraic problems.

Practice Problems

  • Simplify the expression: x³ / (x + 2) - 27 / (x + 2)
  • Simplify the expression: x² / (x - 4) + 16 / (x - 4)
  • Simplify the expression: x⁴ / (x - 1) - 100 / (x - 1)

References