Question

1. factor \\(125x^3 + 27y^6 z^9\\).

\\((5x + 3y^2 z^3)(25x^2 - 15xy^2 z^3 + 9y^4 z^6)\\)

\\((5x + 3y^2 z^3)(25x^2 - 15xy^2 z^3 - 9y^4 z^6)\\)

\\((5x - 3y^2 z^3)(25x^2 + 15xy^2 z^3 - 9y^4 z^6)\\)

\\((5x - 3y^2 z^3)(25x^2 + 15xy^2 z^3 + 9y^4 z^6)\\)

Explanation:

Identify the sum of cubes structure

Using the Exponent Rules and Difference of Cubes knowledge points

$$ 125x^3 + 27y^6z^9 = (5x)^3 + (3y^2z^3)^3 $$

Apply the sum of cubes formula

We apply the algebraic identity for the sum of two cubes:

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

Here, we let \(a = 5x\) and \(b = 3y^2z^3\).

Substitute and simplify terms

Using the Factoring Polynomials knowledge point

$$
LATEXBLOCK0
$$

Write the final factored form

Combining the terms into the sum of cubes formula gives:

$$ (5x + 3y^2z^3)(25x^2 - 15xy^2z^3 + 9y^4z^6) $$

This matches the first option.

Answer:

• (A) \((5x + 3y^2z^3)(25x^2 - 15xy^2z^3 + 9y^4z^6)\) (Correct answer)
• (B) \((5x + 3y^2z^3)(25x^2 - 15xy^2z^3 - 9y^4z^6)\)
• (C) \((5x - 3y^2z^3)(25x^2 + 15xy^2z^3 - 9y^4z^6)\)
• (D) \((5x - 3y^2z^3)(25x^2 + 15xy^2z^3 + 9y^4z^6)\)