To factorize the polynomial [tex]\(p^3 - 343 q^3\)[/tex], we recognize it as a difference of cubes. The general formula for factoring a difference of cubes [tex]\(a^3 - b^3\)[/tex] is:
[tex]\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\][/tex]
Identify [tex]\(a^3 = p^3\)[/tex] and [tex]\(b^3 = 343 q^3\)[/tex]:
[tex]\[p^3 - 343 q^3\][/tex]
Recognize that [tex]\(343 q^3 = (7q)^3\)[/tex], which means:
[tex]\[p^3 - (7q)^3\][/tex]
Now, use the difference of cubes formula with [tex]\(a = p\)[/tex] and [tex]\(b = 7q\)[/tex]:
[tex]\[(p)^3 - (7q)^3 = (p - 7q)\left(p^2 + (p)(7q) + (7q)^2\right)\][/tex]
Expand the terms in the second factor:
[tex]\[
p^2 + (p)(7q) + (7q)^2 = p^2 + 7pq + 49q^2
\][/tex]
Thus, the factorization of the polynomial [tex]\(p^3 - 343 q^3\)[/tex] is:
[tex]\[
(p - 7q)(p^2 + 7pq + 49q^2)
\][/tex]
Among the provided options, the correct factorization corresponds to:
B. [tex]\((p - 7q)(p^2 + 7pq + 49q^2)\)[/tex]
