To expand \((2x - 5)^3\), you can use the formula for the cube of a binomial:

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

For \((2x - 5)^3\), let \(a = 2x\) and \(b = 5\). Applying the formula:

\[
(2x - 5)^3 = (2x)^3 - 3(2x)^2(5) + 3(2x)(5^2) - 5^3
\]

Now, simplify each term:

\[
(2x)^3 = 8x^3
\]
\[
3(2x)^2(5) = 3 \times 4x^2 \times 5 = 60x^2
\]
\[
3(2x)(25) = 3 \times 2x \times 25 = 150x
\]
\[
5^3 = 125
\]

Putting it all together:

\[
(2x - 5)^3 = 8x^3 - 60x^2 + 150x - 125
\]

So, the expanded form of \((2x - 5)^3\) is:

\[
8x^3 - 60x^2 + 150x - 125
\]

Would you like more details or have any questions?

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Here are some related questions you might find helpful:

1. How would you expand \((3x + 4)^3\) step by step?
2. What is the difference between expanding \((a - b)^3\) and \((a + b)^3\)?
3. Can you use the binomial theorem to expand \((2x + 7)^3\)?
4. How would you factor the expression \(8x^3 - 60x^2 + 150x - 125\)?
5. What is the formula for the expansion of \((a + b)^3\)?
6. Can you explain how to simplify each term in the expansion of \((2x - 5)^3\)?
7. How do coefficients change when expanding a binomial with a factor in front of the variable?
8. What are the roots of the cubic equation \(8x^3 - 60x^2 + 150x - 125 = 0\)?

**Tip:** When working with polynomials, always double-check your arithmetic when simplifying each term to avoid errors.

Learn how to expand the cubic polynomial (2x - 5)^3 using the binomial theorem. This detailed step-by-step solution covers the formula for (a - b)^3, simplifying each term, and provides insights into cubic polynomial expansion. Suitable for high school students and beyond.

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