Select All The Correct Answers.Which Expressions Can Be Factored Using The Difference Of Squares Identity?A. 32 Y 2 − 8 Z 2 32y^2 - 8z^2 32 Y 2 − 8 Z 2 B. 6 X 2 − 81 6x^2 - 81 6 X 2 − 81 C. X 4 − 400 X^4 - 400 X 4 − 400 D. 4 A 2 − 64 B 5 4a^2 - 64b^5 4 A 2 − 64 B 5 E. 40 X − X 4 40x - X^4 40 X − X 4

Introduction

In algebra, the difference of squares identity is a fundamental concept used to factorize expressions that can be written in the form of a difference of squares. This identity states that the difference of squares can be factored into the product of two binomials. In this article, we will explore which expressions can be factored using the difference of squares identity.

What is the Difference of Squares Identity?

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 factorize expressions that can be written in the form of a difference of squares.

Which Expressions Can be Factored Using the Difference of Squares Identity?

Let's examine each of the given expressions and determine which ones can be factored using the difference of squares identity.

A. $32y^2 - 8z^2$

This expression can be factored using the difference of squares identity. We can rewrite it as:

$32y^2 - 8z^2 = (16y^2 + 4z^2)(16y^2 - 4z^2)$

However, we can simplify it further by factoring out the greatest common factor (GCF) of 16:

$32y^2 - 8z^2 = 16(2y^2 - z^2)$

Now, we can apply the difference of squares identity to the expression inside the parentheses:

$2y^2 - z^2 = (y + z)(y - z)$

Therefore, the final factored form of the expression is:

$32y^2 - 8z^2 = 16(y + z)(y - z)$

B. $6x^2 - 81$

This expression cannot be factored using the difference of squares identity. The reason is that the expression is not in the form of a difference of squares. However, we can factor out the greatest common factor (GCF) of 3:

$6x^2 - 81 = 3(2x^2 - 27)$

Now, we can rewrite the expression as:

$2x^2 - 27 = 2x^2 - 9^2$

However, this is still not in the form of a difference of squares. Therefore, the expression $6x^2 - 81$ cannot be factored using the difference of squares identity.

C. $x^4 - 400$

This expression can be factored using the difference of squares identity. We can rewrite it as:

$x^4 - 400 = x^4 - 20^2$

Now, we can apply the difference of squares identity to the expression:

$x^4 - 20^2 = (x^2 + 20)(x^2 - 20)$

However, we can simplify it further by factoring out the greatest common factor (GCF) of $x^2$:

$x^4 - 400 = x^2(x^2 + 20)(x^2 - 20)$

Now, we can apply the difference of squares identity to the expression inside the parentheses:

$x^2 - 20 = (x + \sqrt{20})(x - \sqrt{20})$

Therefore, the final factored form of the expression is:

$x^4 - 400 = x^2(x + \sqrt{20})(x - \sqrt{20})(x^2 + 20)$

D. $4a^2 - 64b^5$

This expression can be factored using the difference of squares identity. We can rewrite it as:

$4a^2 - 64b^5 = 4a^2 - (4b^5)^2$

Now, we can apply the difference of squares identity to the expression:

$4a^2 - (4b^5)^2 = (2a + 4b^5)(2a - 4b^5)$

However, we can simplify it further by factoring out the greatest common factor (GCF) of 2:

$4a^2 - 64b^5 = 2(2a + 4b^5)(2a - 4b^5)$

Therefore, the final factored form of the expression is:

$4a^2 - 64b^5 = 2(2a + 4b^5)(2a - 4b^5)$

E. $40x - x^4$

This expression cannot be factored using the difference of squares identity. The reason is that the expression is not in the form of a difference of squares. However, we can rewrite it as:

$40x - x^4 = -x^4 + 40x$

Now, we can rewrite the expression as:

$-x^4 + 40x = -x^4 + 4(10)x$

However, this is still not in the form of a difference of squares. Therefore, the expression $40x - x^4$ cannot be factored using the difference of squares identity.

Conclusion

Introduction

In our previous article, we explored which expressions can be factored using the difference of squares identity. In this article, we will answer some frequently asked questions about factoring expressions using this 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 factorize expressions that can be written in the form of a difference of squares.

Q: How do I know if an expression can be factored using the difference of squares identity?

A: To determine if an expression can be factored using the difference of squares identity, you need to check if it can be written in the form of a difference of squares. This means that the expression must be in the form of a^2 - b^2, where a and b are expressions.

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 checking if the expression can be written in the form of a difference of squares before applying the identity.
• Not factoring out the greatest common factor (GCF) before applying the identity.
• Not simplifying the expression after applying the identity.

Q: Can the difference of squares identity be used to factor expressions that are not in the form of a difference of squares?

A: No, the difference of squares identity can only be used to factor expressions that are in the form of a difference of squares. If an expression is not in this form, it cannot be factored using this identity.

Q: How do I factor expressions that are in the form of a difference of squares but have a coefficient other than 1?

A: To factor expressions that are in the form of a difference of squares but have a coefficient other than 1, you need to factor out the coefficient before applying the difference of squares identity. For example, if you have the expression 3x^2 - 9, you can factor out the coefficient 3 before applying the identity:

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

Now, you can apply the difference of squares identity to the expression inside the parentheses:

x^2 - 3 = (x + \sqrt{3})(x - \sqrt{3})

Therefore, the final factored form of the expression is:

3x^2 - 9 = 3(x + \sqrt{3})(x - \sqrt{3})

Q: Can the difference of squares identity be used to factor expressions that have a variable in the denominator?

A: No, the difference of squares identity cannot be used to factor expressions that have a variable in the denominator. This is because the identity requires that the expressions be in the form of a difference of squares, and expressions with variables in the denominator are not in this form.

Q: How do I determine if an expression can be factored using the difference of squares identity if it has a variable in the denominator?

A: To determine if an expression can be factored using the difference of squares identity if it has a variable in the denominator, you need to check if the expression can be written in the form of a difference of squares. If the expression has a variable in the denominator, it is unlikely to be in this form, and the difference of squares identity cannot be used to factor it.

Conclusion

In this article, we have answered some frequently asked questions about factoring expressions using the difference of squares identity. We have seen that this identity can be used to factor expressions that are in the form of a difference of squares, but it cannot be used to factor expressions that are not in this form. By understanding the difference of squares identity and how to apply it, you can simplify complex expressions and make them easier to work with.