Factor Completely $16x^4 - 81$.A. $(2x - 3)(2x - 3)(4x^2 + 9)$ B. \$(2x - 3)(2x + 3)(4x^2 + 9)$[/tex\] C. $(2x - 3)(2x + 3)(4x^2 - 9)$ D. $(2x + 3)(2x + 3)(4x^2 + 9)$

Introduction

In algebra, factoring is a crucial concept that helps us simplify complex expressions and solve equations. One of the most common types of factoring is the difference of squares, which is a fundamental concept in mathematics. In this article, we will explore how to factor the difference of squares, using the given problem as an example.

The Problem

The problem asks us to factor the expression $16x^4 - 81$. To factor this expression, we need to identify the difference of squares pattern.

Understanding the Difference of Squares

The difference of squares is a mathematical formula that states:

$$a^2 - b^2 = (a + b)(a - b)$$

This formula can be applied to any two numbers or expressions, as long as they are squared. In our problem, we have:

$$16x^4 - 81$$

We can rewrite this expression as:

$$(4x^2)^2 - 9^2$$

Now, we can see that we have a difference of squares, where $a = 4x^2$ and $b = 9$.

Factoring the Difference of Squares

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

$$(4x^2 + 9)(4x^2 - 9)$$

However, we are not done yet. We can further factor the expression by recognizing that $4x^2 - 9$ is also a difference of squares.

Factoring the Second Difference of Squares

We can rewrite $4x^2 - 9$ as:

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

Now, we can apply the difference of squares formula again:

$$(2x + 3)(2x - 3)$$

Combining the Factors

We can now combine the factors we have obtained so far:

$$(4x^2 + 9)(2x + 3)(2x - 3)$$

However, we can simplify this expression further by recognizing that $4x^2 + 9$ is a sum of squares, which cannot be factored further.

The Final Answer

Therefore, the final answer is:

$$(2x - 3)(2x + 3)(4x^2 + 9)$$

Conclusion

In this article, we have explored how to factor the difference of squares, using the given problem as an example. We have seen how to recognize the difference of squares pattern, apply the formula, and combine the factors to obtain the final answer. This concept is crucial in algebra and is used extensively in mathematics and other fields.

Common Mistakes to Avoid

When factoring the difference of squares, it is essential to recognize the pattern and apply the formula correctly. Some common mistakes to avoid include:

• Not recognizing the difference of squares pattern
• Applying the formula incorrectly
• Not combining the factors correctly

Tips and Tricks

To factor the difference of squares, follow these tips and tricks:

• Look for the difference of squares pattern
• Apply the formula correctly
• Combine the factors carefully
• Check your work to ensure that the expression is factored correctly

Real-World Applications

The concept of factoring the difference of squares has numerous real-world applications, including:

• Simplifying complex expressions
• Solving equations
• Analyzing data
• Making predictions

Conclusion

Introduction

In our previous article, we explored how to factor the difference of squares, using the given problem as an example. In this article, we will answer some of the most frequently asked questions about factoring the difference of squares.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

$$a^2 - b^2 = (a + b)(a - b)$$

This formula can be applied to any two numbers or expressions, as long as they are squared.

Q: How do I recognize the difference of squares pattern?

A: To recognize the difference of squares pattern, look for two squared expressions that are being subtracted. For example:

$$16x^4 - 81$$

We can rewrite this expression as:

$$(4x^2)^2 - 9^2$$

Now, we can see that we have a difference of squares, where $a = 4x^2$ and $b = 9$.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, simply substitute the values of $a$ and $b$ into the formula:

$$(a + b)(a - b)$$

In our example, we have:

$$(4x^2 + 9)(4x^2 - 9)$$

Q: Can I factor the difference of squares further?

A: Yes, you can factor the difference of squares further by recognizing that $4x^2 - 9$ is also a difference of squares:

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

Now, we can apply the difference of squares formula again:

$$(2x + 3)(2x - 3)$$

Q: How do I combine the factors?

A: To combine the factors, simply multiply the expressions together:

$$(4x^2 + 9)(2x + 3)(2x - 3)$$

However, we can simplify this expression further by recognizing that $4x^2 + 9$ is a sum of squares, which cannot be factored further.

Q: What are some common mistakes to avoid when factoring the difference of squares?

A: Some common mistakes to avoid when factoring the difference of squares include:

• Not recognizing the difference of squares pattern
• Applying the formula incorrectly
• Not combining the factors correctly

Q: What are some real-world applications of factoring the difference of squares?

A: The concept of factoring the difference of squares has numerous real-world applications, including:

• Simplifying complex expressions
• Solving equations
• Analyzing data
• Making predictions

Q: How can I practice factoring the difference of squares?

A: To practice factoring the difference of squares, try the following:

• Start with simple expressions, such as $x^2 - 4$
• Gradually move on to more complex expressions, such as $16x^4 - 81$
• Use online resources, such as math websites and video tutorials, to help you practice
• Work with a partner or tutor to get feedback and guidance

Conclusion

In conclusion, factoring the difference of squares is a crucial concept in mathematics that has numerous real-world applications. By recognizing the pattern, applying the formula, and combining the factors correctly, we can simplify complex expressions and solve equations. This concept is essential in algebra and is used extensively in mathematics and other fields.