Question

1. factor \\(8m^6 - 64n^{12}p^{15}\\). which of the following is one of the factors?

\\((2m^2 - 4n^4p^5)\\)

\\((2m^2 + 4n^4p^5)\\)

\\((4m^4 + 8m^2n^4p^5 + 16n^{16}p^{25})\\)

\\((4m^4 - 8m^2n^4p^5 + 16n^8p^{10})\\)

Explanation:

Identify the structure as a difference of cubes

Using the Exponent Rules and Algebraic Simplification knowledge points

$$ 8m^6 - 64n^{12}p^{15} = (2m^2)^3 - (4n^4p^5)^3 $$

Apply the difference of cubes formula

Using the Algebraic Simplification knowledge point

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

$$ \text{Let } a = 2m^2 \text{ and } b = 4n^4p^5 $$

Determine the binomial factor

Using the Algebraic Simplification knowledge point

$$ a - b = 2m^2 - 4n^4p^5 $$

Determine the trinomial factor

Using the Exponent Rules and Algebraic Simplification knowledge points

$$
LATEXBLOCK0
$$

Match with the given options

Using the Algebraic Simplification knowledge point

$$ \text{The factors are } (2m^2 - 4n^4p^5) \text{ and } (4m^4 + 8m^2n^4p^5 + 16n^8p^{10}) $$

$$ \text{The option } (2m^2 - 4n^4p^5) \text{ is listed as the first choice.} $$

Answer:

• (A) \((2m^2 - 4n^4p^5)\) (Correct answer)
• (B) \((2m^2 + 4n^4p^5)\)
• (C) \((4m^4 + 8m^2n^4p^5 + 16n^{16}p^{25})\)
• (D) \((4m^4 - 8m^2n^4p^5 + 16n^8p^{10})\)