Solve the polynomial using either the sum of perfect cubes or difference of perfect cubes. Find at least one solution, real or complex, and separate multiple answers with a comma.

512
t
3
−
729
=
0
512t 
3
 −729=0

The given equation is:

\[
512t^3 - 729 = 0
\]

This is a difference of cubes, which follows the general form:

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

First, rewrite the equation in terms of cubes:

\[
(8t)^3 - 9^3 = 0
\]

So, \(a = 8t\) and \(b = 9\).

Using the difference of cubes formula:

\[
(8t - 9)((8t)^2 + (8t)(9) + 9^2) = 0
\]

Now, simplify each factor:

1. \( 8t - 9 \)
2. \( (8t)^2 + (8t)(9) + 9^2 = 64t^2 + 72t + 81 \)

The equation becomes:

\[
(8t - 9)(64t^2 + 72t + 81) = 0
\]

### Finding solutions:
1. From the first factor: 
\[
8t - 9 = 0 \implies t = \frac{9}{8}
\]

2. For the quadratic factor \(64t^2 + 72t + 81 = 0\), solve using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \(a = 64\), \(b = 72\), and \(c = 81\). Calculate the discriminant:

\[
\Delta = b^2 - 4ac = 72^2 - 4(64)(81) = 5184 - 20736 = -15552
\]

Since the discriminant is negative, the quadratic has complex solutions. The solutions are:

\[
t = \frac{-72 \pm \sqrt{-15552}}{2(64)} = \frac{-72 \pm 124.7i}{128}
\]

Simplifying:

\[
t = \frac{-72}{128} \pm \frac{124.7i}{128} = -\frac{9}{16} \pm \frac{124.7i}{128}
\]

Thus, the complex solutions are:

\[
t = -\frac{9}{16} \pm \frac{124.7i}{128}
\]

### Final answer:
The solutions are:
\[
t = \frac{9}{8}, -\frac{9}{16} \pm \frac{124.7i}{128}
\]

Would you like more details or have any questions?

Here are five related questions for deeper understanding:
1. How do you use the difference of cubes to solve polynomials?
2. What is the importance of the discriminant in determining solution types?
3. How would you solve a sum of cubes equation?
4. Why do complex numbers appear when the discriminant is negative?
5. Can this method be applied to higher-degree polynomials?

**Tip:** Always check for real and complex solutions using the discriminant to predict the nature of the roots!

Solve the polynomial using either the sum of perfect cubes or difference of perfect cubes. Find at least one solution, real or complex, and separate multiple answers with a comma.

512t^3 − 729 = 0

Learn how to solve the polynomial 512t^3 − 729 using the difference of cubes formula. This problem explores key algebraic concepts like polynomials, the difference of cubes theorem, and solving quadratic equations with complex solutions. Suitable for high school students in Grades 10-12.

Math Problem Statement

512 t 3 − 729

Solution

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