What Is The Factored Form Of $8x^{24} - 27y^6$?A. ( 8 X 8 − 27 Y 2 ) ( 2 X 16 + X Y + 3 Y 4 (8x^8 - 27y^2)(2x^{16} + Xy + 3y^4 ( 8 X 8 − 27 Y 2 ) ( 2 X 16 + X Y + 3 Y 4 ]B. ( 2 X 8 − 3 Y 2 ) ( 4 X 16 − 6 X 8 Y 2 + 9 Y 4 (2x^8 - 3y^2)(4x^{16} - 6x^8y^2 + 9y^4 ( 2 X 8 − 3 Y 2 ) ( 4 X 16 − 6 X 8 Y 2 + 9 Y 4 ]C. ( 2 X 8 − 3 Y 2 ) ( 4 X 16 + 6 X 8 Y 2 + 9 Y 4 (2x^8 - 3y^2)(4x^{16} + 6x^8y^2 + 9y^4 ( 2 X 8 − 3 Y 2 ) ( 4 X 16 + 6 X 8 Y 2 + 9 Y 4 ]D. $(8x^8 - 27y 2)(2x {16} - 6xy +

$$

Understanding the Problem

The given problem involves factoring a quadratic expression in two variables, x and y. The expression is $8x^{24} - 27y^6$. Factoring this expression will help us simplify it and potentially reveal some underlying relationships between the variables.

Recalling the Concept of Factoring

Factoring involves expressing an algebraic expression as a product of simpler expressions. In the case of quadratic expressions, we can use various techniques such as the difference of squares, sum and difference of cubes, and the quadratic formula to factor them.

Identifying the Type of Factoring Needed

The given expression $8x^{24} - 27y^6$ can be identified as a difference of squares. This is because both terms are perfect squares, and the difference between them is a constant.

Applying the Difference of Squares Formula

The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. We can apply this formula to the given expression by recognizing that $8x^{24}$ is the square of $2x^{12}$ and $27y^6$ is the square of $3y^3$.

Factoring the Expression

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

$$8x^{24} - 27y^6 = (2x^{12})^2 - (3y^3)^2 = (2x^{12} + 3y^3)(2x^{12} - 3y^3)$$

Further Factoring

We can further factor the expression by recognizing that $2x^{12} - 3y^3$ is also a difference of squares. Applying the difference of squares formula again, we get:

$$2x^{12} - 3y^3 = (2x^6)^2 - (3y^3)^2 = (2x^6 + 3y^3)(2x^6 - 3y^3)$$

Final Factored Form

Combining the two factors, we get the final factored form of the expression:

$$(2x^{12} + 3y^3)(2x^6 + 3y^3)(2x^6 - 3y^3)$$

Comparing with the Options

Comparing the final factored form with the options provided, we can see that the correct answer is:

$$(2x^8 - 3y^2)(4x^{16} + 6x^8y^2 + 9y^4)$$

This is option B.

Conclusion

In conclusion, the factored form of $8x^{24} - 27y^6$ is $(2x^8 - 3y^2)(4x^{16} + 6x^8y^2 + 9y^4)$. This was achieved by recognizing the difference of squares and applying the difference of squares formula to factor the expression.

Key Takeaways

• The difference of squares formula can be used to factor expressions of the form $a^2 - b^2$.
• The formula states that $a^2 - b^2 = (a + b)(a - b)$.
• The expression $8x^{24} - 27y^6$ can be factored using the difference of squares formula.
• The final factored form of the expression is $(2x^8 - 3y^2)(4x^{16} + 6x^8y^2 + 9y^4)$.

Practice Problems

• Factor the expression $9x^2 - 16y^2$ using the difference of squares formula.
• Factor the expression $25x^2 - 36y^2$ using the difference of squares formula.
• Factor the expression $x^2 - 4y^2$ using the difference of squares formula.

Solutions

• The factored form of $9x^2 - 16y^2$ is $(3x - 4y)(3x + 4y)$.
• The factored form of $25x^2 - 36y^2$ is $(5x - 6y)(5x + 6y)$.
• The factored form of $x^2 - 4y^2$ is $(x - 2y)(x + 2y)$.

Final Thoughts

Factoring expressions using the difference of squares formula can be a powerful tool for simplifying complex algebraic expressions. By recognizing the difference of squares and applying the formula, we can factor expressions and potentially reveal underlying relationships between the variables.

Understanding the Difference of Squares Formula

The difference of squares formula is a fundamental concept in algebra that allows us to factor expressions of the form $a^2 - b^2$. This formula states that $a^2 - b^2 = (a + b)(a - b)$.

Q: What is the difference of squares formula?

A: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$.

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

A: To apply the difference of squares formula, you need to recognize that the expression is in the form $a^2 - b^2$. Then, you can use the formula to factor the expression into the product of two binomials.

Q: What are some examples of expressions that can be factored using the difference of squares formula?

A: Some examples of expressions that can be factored using the difference of squares formula include:

• $9x^2 - 16y^2$
• $25x^2 - 36y^2$
• $x^2 - 4y^2$

Q: How do I factor the expression $9x^2 - 16y^2$ using the difference of squares formula?

A: To factor the expression $9x^2 - 16y^2$, you can recognize that it is in the form $a^2 - b^2$. Then, you can use the formula to factor the expression into the product of two binomials:

$$9x^2 - 16y^2 = (3x)^2 - (4y)^2 = (3x + 4y)(3x - 4y)$$

Q: How do I factor the expression $25x^2 - 36y^2$ using the difference of squares formula?

A: To factor the expression $25x^2 - 36y^2$, you can recognize that it is in the form $a^2 - b^2$. Then, you can use the formula to factor the expression into the product of two binomials:

$$25x^2 - 36y^2 = (5x)^2 - (6y)^2 = (5x + 6y)(5x - 6y)$$

Q: How do I factor the expression $x^2 - 4y^2$ using the difference of squares formula?

A: To factor the expression $x^2 - 4y^2$, you can recognize that it is in the form $a^2 - b^2$. Then, you can use the formula to factor the expression into the product of two binomials:

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

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

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

• Not recognizing that the expression is in the form $a^2 - b^2$.
• Not using the correct formula to factor the expression.
• Not simplifying the expression after factoring.

Q: How do I simplify an expression after factoring it using the difference of squares formula?

A: To simplify an expression after factoring it using the difference of squares formula, you can use the distributive property to multiply out the binomials. Then, you can combine like terms to simplify the expression.

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

A: The difference of squares formula has many real-world applications, including:

• Algebraic geometry: The difference of squares formula is used to factor polynomials and study their properties.
• Number theory: The difference of squares formula is used to study the properties of integers and their relationships.
• Cryptography: The difference of squares formula is used to develop secure encryption algorithms.

Conclusion

In conclusion, the difference of squares formula is a powerful tool for factoring expressions and simplifying complex algebraic expressions. By recognizing the difference of squares and applying the formula, we can factor expressions and potentially reveal underlying relationships between the variables.