Factorizing 1 - 64z^3: A Step-by-Step Guide

Factoring algebraic expressions is a foundational skill in algebra. One common type of expression that often appears is the difference of cubes. In this article, we will walk through the process of factorizing the expression 1 - 64z^3, providing a clear and detailed explanation.

Understanding the Difference of Cubes

The difference of cubes is a special algebraic identity that helps us factor expressions of the form a^3 - b^3. The identity states:

a^3 - b^3 (a - b)(a^2 ab b^2)

This identity makes use of the fact that cubes of numbers can be factored into simpler parts, and it is extremely useful in solving various algebraic problems.

Identifying the Components

To apply the difference of cubes formula to the expression 1 - 64z^3, we first need to identify the values of a and b that fit into the formula. In this case, we can see that:

Now that we have identified a and b, we can proceed to apply the formula.

Applying the Difference of Cubes Formula

Next, let's apply the difference of cubes formula to 1 - 64z^3. According to the formula:

a^3 - b^3 (a - b)(a^2 ab b^2)

Substituting a 1 and b 4z into the formula, we get:

1 - 64z^3 (1 - 4z)((1)^2 (1)(4z) (4z)^2)

Now we need to simplify each part of the expression:

Substituting these simplified parts back into the formula, we obtain:

1 - 64z^3 (1 - 4z)(1 4z 16z^2)

This is the fully factored form of the expression 1 - 64z^3.

Conclusion

By recognizing the expression 1 - 64z^3 as a difference of cubes and applying the appropriate formula, we were able to factor it successfully. Understanding and applying the difference of cubes formula is a powerful tool in algebra, and it can be applied to a wide range of problems involving polynomial factorization.

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