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Introduction

In algebra, there are several important identities that help us simplify complex expressions and solve equations. One of the most useful identities is the difference of two cubes, which states that:

$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

This identity can be used to factor expressions of the form $a^3 - b^3$, where $a$ and $b$ are any real numbers. In this article, we will use this identity to factor the expression $64x^6 - 27$.

The Difference of Two Cubes Identity

The difference of two cubes identity is a fundamental concept in algebra that can be used to simplify complex expressions. The identity states that:

$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

This identity can be proven using the following steps:

1. Factor the left-hand side of the equation as a difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
2. Expand the right-hand side of the equation: $(a - b)(a^2 + ab + b^2) = a^3 - b^3$
3. Simplify the equation: $a^3 - b^3 = a^3 - b^3$

This identity can be used to factor expressions of the form $a^3 - b^3$, where $a$ and $b$ are any real numbers.

Factoring the Expression $64x^6 - 27$

Now that we have the difference of two cubes identity, we can use it to factor the expression $64x^6 - 27$. To do this, we need to identify the values of $a$ and $b$ in the expression.

In this case, we can see that $a = 4x^2$ and $b = 3$. Therefore, we can write the expression as:

$$64x^6 - 27 = (4x^2)^3 - 3^3$$

Now we can use the difference of two cubes identity to factor the expression:

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

Simplifying the expression, we get:

$$(4x^2 - 3)(16x^4 + 12x^2 + 9)$$

Therefore, the factored form of the expression $64x^6 - 27$ is:

$$(4x^2 - 3)(16x^4 + 12x^2 + 9)$$

Conclusion

In this article, we used the difference of two cubes identity to factor the expression $64x^6 - 27$. We identified the values of $a$ and $b$ in the expression and used the identity to simplify the expression. The factored form of the expression is $(4x^2 - 3)(16x^4 + 12x^2 + 9)$.

Example Problems

Here are some example problems that you can use to practice using the difference of two cubes identity:

1. Factor the expression $27x^6 - 8$.
2. Factor the expression $64x^6 - 125$.
3. Factor the expression $27x^6 - 27$.

Solutions

1. Factor the expression $27x^6 - 8$.

Using the difference of two cubes identity, we can write the expression as:

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

Simplifying the expression, we get:

$$(3x^2 - 2)(9x^4 + 6x^2 + 4)$$

2. Factor the expression $64x^6 - 125$.

Using the difference of two cubes identity, we can write the expression as:

$$(4x^2)^3 - 5^3 = (4x^2 - 5)((4x^2)^2 + (4x^2)(5) + 5^2)$$

Simplifying the expression, we get:

$$(4x^2 - 5)(16x^4 + 20x^2 + 25)$$

3. Factor the expression $27x^6 - 27$.

Using the difference of two cubes identity, we can write the expression as:

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

Simplifying the expression, we get:

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

Tips and Tricks

Here are some tips and tricks that you can use to help you factor expressions using the difference of two cubes identity:

1. Identify the values of $a$ and $b$ in the expression.
2. Use the difference of two cubes identity to factor the expression.
3. Simplify the expression by combining like terms.
4. Check your work by plugging the factored expression back into the original expression.

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the difference of two cubes identity.

Q: What is the difference of two cubes identity?

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

$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

This identity can be used to factor expressions of the form $a^3 - b^3$, where $a$ and $b$ are any real numbers.

Q: How do I use the difference of two cubes identity to factor an expression?

A: To use the difference of two cubes identity to factor an expression, follow these steps:

1. Identify the values of $a$ and $b$ in the expression.
2. Use the difference of two cubes identity to factor the expression.
3. Simplify the expression by combining like terms.

Q: What are some common mistakes to avoid when using the difference of two cubes identity?

A: Here are some common mistakes to avoid when using the difference of two cubes identity:

1. Not identifying the values of $a$ and $b$ in the expression.
2. Not using the correct formula to factor the expression.
3. Not simplifying the expression by combining like terms.

Q: Can I use the difference of two cubes identity to factor expressions that are not in the form $a^3 - b^3$?

A: No, the difference of two cubes identity can only be used to factor expressions that are in the form $a^3 - b^3$. If you have an expression that is not in this form, you will need to use a different method to factor it.

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

A: To determine if an expression can be factored using the difference of two cubes identity, look for the following:

1. The expression must be in the form $a^3 - b^3$.
2. The expression must have two terms that are cubes.

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

A: Yes, you can use the difference of two cubes identity to factor expressions with negative numbers. However, you will need to be careful when simplifying the expression, as the negative sign may affect the final result.

Q: How do I simplify an expression that has been factored using the difference of two cubes identity?

A: To simplify an expression that has been factored using the difference of two cubes identity, follow these steps:

1. Combine like terms.
2. Simplify any remaining expressions.

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

A: Yes, you can use the difference of two cubes identity to factor expressions with variables. However, you will need to be careful when simplifying the expression, as the variable may affect the final result.

Q: How do I know if I have factored an expression correctly using the difference of two cubes identity?

A: To determine if you have factored an expression correctly using the difference of two cubes identity, follow these steps:

1. Check that the expression is in the form $a^3 - b^3$.
2. Check that the expression has been factored correctly using the difference of two cubes identity.
3. Check that the simplified expression is correct.

By following these steps, you can ensure that you have factored an expression correctly using the difference of two cubes identity.

Common Mistakes

Here are some common mistakes to avoid when using the difference of two cubes identity:

1. Not identifying the values of $a$ and $b$ in the expression: Make sure to identify the values of $a$ and $b$ in the expression before using the difference of two cubes identity.
2. Not using the correct formula to factor the expression: Make sure to use the correct formula to factor the expression.
3. Not simplifying the expression by combining like terms: Make sure to simplify the expression by combining like terms.

Tips and Tricks

Here are some tips and tricks to help you use the difference of two cubes identity effectively:

1. Practice, practice, practice: The more you practice using the difference of two cubes identity, the more comfortable you will become with it.
2. Use the difference of two cubes identity to factor expressions with small numbers: Start by using the difference of two cubes identity to factor expressions with small numbers, such as $a^3 - b^3$.
3. Use the difference of two cubes identity to factor expressions with variables: Once you are comfortable using the difference of two cubes identity to factor expressions with small numbers, try using it to factor expressions with variables.

By following these tips and tricks, you can become proficient in using the difference of two cubes identity to factor expressions.