Which Expressions Can Be Factored Using The Difference Of Squares Identity?A. ${ 6x^2 - 81\$} B. { X^4 - 400$}$C. ${ 32y^2 - 8z^2\$} D. ${ 49x - X^4\$} E. ${ 4a^2 - 64b^5\$}

Introduction

In algebra, factoring expressions is a crucial skill that helps us simplify complex equations and solve problems more efficiently. One of the most useful techniques for factoring is the difference of squares identity, which states that:

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

This identity can be used to factor expressions that have a difference of squares, which is a common pattern in algebra. In this article, we will explore which expressions can be factored using the difference of squares identity.

Understanding the Difference of Squares Identity

Before we dive into the examples, let's make sure we understand the difference of squares identity. The identity states that the difference of two squares can be factored into the product of two binomials. The binomials are:

• (a + b): This is the sum of the two squares.
• (a - b): This is the difference of the two squares.

To use the difference of squares identity, we need to identify the two squares in the expression. The two squares are the terms that are being subtracted.

Example 1: Factoring 6x^2 - 81

Let's start with the first expression: 6x^2 - 81. To factor this expression using the difference of squares identity, we need to identify the two squares. In this case, the two squares are 6x^2 and 81.

We can rewrite 6x^2 as (2x)^2 and 81 as 9^2. Now we can see that the expression is in the form of a difference of squares:

(2x)^2 - 9^2

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

(2x + 9)(2x - 9)

Therefore, the correct answer is A. 6x^2 - 81.

Example 2: Factoring x^4 - 400

The second expression is x^4 - 400. To factor this expression using the difference of squares identity, we need to identify the two squares. In this case, the two squares are x^4 and 400.

We can rewrite x^4 as (x²⁾2 and 400 as 20^2. Now we can see that the expression is in the form of a difference of squares:

(x²⁾2 - 20^2

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

(x^2 + 20)(x^2 - 20)

However, we can further factor the expression (x^2 - 20) as:

(x + sqrt(20))(x - sqrt(20))

Therefore, the correct answer is B. x^4 - 400.

Example 3: Factoring 32y^2 - 8z^2

The third expression is 32y^2 - 8z^2. To factor this expression using the difference of squares identity, we need to identify the two squares. In this case, the two squares are 32y^2 and 8z^2.

We can rewrite 32y^2 as (4y)^2 and 8z^2 as (2z)^2. Now we can see that the expression is in the form of a difference of squares:

(4y)^2 - (2z)^2

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

(4y + 2z)(4y - 2z)

Therefore, the correct answer is C. 32y^2 - 8z^2.

Example 4: Factoring 49x - x^4

The fourth expression is 49x - x^4. To factor this expression using the difference of squares identity, we need to identify the two squares. In this case, the two squares are 49x and x^4.

We can rewrite 49x as (7x)^2 and x^4 as (x²⁾2. Now we can see that the expression is in the form of a difference of squares:

(7x)^2 - (x²⁾2

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

(7x + x^2)(7x - x^2)

However, we can further factor the expression (7x - x^2) as:

(x(7 - x))

Therefore, the correct answer is D. 49x - x^4.

Example 5: Factoring 4a^2 - 64b^5

The fifth expression is 4a^2 - 64b^5. To factor this expression using the difference of squares identity, we need to identify the two squares. In this case, the two squares are 4a^2 and 64b^5.

We can rewrite 4a^2 as (2a)^2 and 64b^5 as (8b²⁾2. Now we can see that the expression is in the form of a difference of squares:

(2a)^2 - (8b²⁾2

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

(2a + 8b^2)(2a - 8b^2)

However, we can further factor the expression (2a - 8b^2) as:

(2(a - 4b^2))

Therefore, the correct answer is E. 4a^2 - 64b^5.

Conclusion

In conclusion, the difference of squares identity is a powerful tool for factoring expressions. By identifying the two squares in the expression, we can use the difference of squares identity to factor the expression into the product of two binomials. In this article, we have explored five expressions and determined which ones can be factored using the difference of squares identity.

The correct answers are:

• A. 6x^2 - 81
• B. x^4 - 400
• C. 32y^2 - 8z^2
• D. 49x - x^4
• E. 4a^2 - 64b^5

Introduction

In our previous article, we explored the difference of squares identity and how to use it to factor expressions. In this article, we will answer some common questions about factoring expressions using the difference of squares identity.

Q: What is the difference of squares identity?

A: The difference of squares identity is a mathematical formula that states:

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

This identity can be used to factor expressions that have a difference of squares.

Q: How do I identify the two squares in an expression?

A: To identify the two squares in an expression, you need to look for terms that are being subtracted. The two squares are the terms that are being subtracted.

For example, in the expression 6x^2 - 81, the two squares are 6x^2 and 81.

Q: How do I factor an expression using the difference of squares identity?

A: To factor an expression using the difference of squares identity, you need to follow these steps:

1. Identify the two squares in the expression.
2. Rewrite the expression as a difference of squares.
3. Use the difference of squares identity to factor the expression.

For example, in the expression 6x^2 - 81, we can rewrite it as:

(2x)^2 - 9^2

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

(2x + 9)(2x - 9)

Q: Can I use the difference of squares identity to factor expressions with variables?

A: Yes, you can use the difference of squares identity to factor expressions with variables. For example, in the expression x^4 - 400, we can rewrite it as:

(x²⁾2 - 20^2

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

(x^2 + 20)(x^2 - 20)

Q: Can I use the difference of squares identity to factor expressions with negative numbers?

A: Yes, you can use the difference of squares identity to factor expressions with negative numbers. For example, in the expression -32y^2 + 8z^2, we can rewrite it as:

(-4y)^2 - (2z)^2

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

(-4y + 2z)(-4y - 2z)

Q: Can I use the difference of squares identity to factor expressions with fractions?

A: Yes, you can use the difference of squares identity to factor expressions with fractions. For example, in the expression (1/2)^2 - (1/4)^2, we can rewrite it as:

(1/2 + 1/4)(1/2 - 1/4)

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

(3/4)(1/4)

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

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

• Not identifying the two squares in the expression.
• Not rewriting the expression as a difference of squares.
• Not using the correct signs when factoring the expression.

Conclusion

In conclusion, the difference of squares identity is a powerful tool for factoring expressions. By identifying the two squares in the expression and using the difference of squares identity, you can factor expressions into the product of two binomials. We hope this article has been helpful in answering some common questions about factoring expressions using the difference of squares identity.

Additional Resources

If you need additional help with factoring expressions using the difference of squares identity, here are some additional resources:

• Khan Academy: Factoring Expressions Using the Difference of Squares Identity
• Mathway: Factoring Expressions Using the Difference of Squares Identity
• Wolfram Alpha: Factoring Expressions Using the Difference of Squares Identity

We hope this article has been helpful in understanding the difference of squares identity and how to use it to factor expressions.