Factor using the formula for the sum or difference of two cubes.
729 x minus x Superscript 4
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Part 1
729 x minus x Superscript 4equals
  
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To factor the expression \( 729x - x^4 \), we can start by rewriting it to fit the form of a difference of cubes. Notice that \( 729x = (9x)^3 \) and \( x^4 = (x^2)^2 \). Therefore, the expression becomes:

\[
729x - x^4 = (9x)^3 - (x^2)^2
\]

Now, let's apply the difference of cubes formula, which states:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

In our case, let:
- \( a = 9x \)
- \( b = x^2 \)

Substitute these values into the difference of cubes formula:

\[
(9x)^3 - (x^2)^3 = (9x - x^2)((9x)^2 + 9x \cdot x^2 + (x^2)^2)
\]

Simplify each term in the second factor:

\[
(9x)^2 = 81x^2, \quad 9x \cdot x^2 = 9x^3, \quad (x^2)^2 = x^4
\]

Thus, the factored form is:

\[
(9x - x^2)(81x^2 + 9x^3 + x^4)
\]

This is the fully factored expression of \( 729x - x^4 \).

Would you like more details on any of the steps?

Here are 5 follow-up questions for further exploration:
1. Can you apply the sum of cubes formula to a different expression?
2. How does the difference of squares formula differ from the difference of cubes?
3. What happens if the expression is a sum of cubes instead of a difference?
4. Can you factor higher powers using a similar approach?
5. What is the process for factoring when the coefficients are not perfect cubes?

**Tip:** Always try to rewrite expressions in terms of perfect powers (like cubes or squares) before applying formulas.

Factor using the formula for the sum or difference of two cubes. 729x minus x^4

Learn how to factor the expression 729x - x^4 using the difference of cubes formula. This step-by-step guide explains the process, including identifying cubes and applying the formula to factor the polynomial. Suitable for high school students in Grades 9-12.

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