Question

Which of the following is the correct factorization of the polynomial p^3 - 34303?

A) (p-499)(p^2 + 7pq + 4902)
B) (p-79)(p^2 + 7pq + 49)
C) (p^2 + 79)(p^3 + 49pq + 792)
D) The polynomial is irreducible.

Answer 1

The correct factorization of the polynomial is (p - 7)(p^2 + 7p + 49), obtained by recognizing it as a difference of cubes and applying the formula for such a difference. None of the provided options is correct, but Option B is the closest with a typo in the first term.

Step-by-step explanation:

The correct factorization of the polynomial p^3 - 343 can be found by recognizing it as a difference of cubes. A difference of cubes is factored using the formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2). To apply this to the given polynomial, we must identify the cube root of 343, which is 7 since 7x7x7=343.

Using the formula, p^3 - 343 can be rewritten as p^3 - 7^3. This gives us the factorization (p - 7)(p^2 + 7p + 49). None of the provided options matches this factorization exactly, suggesting a potential typo in the question. However, Option B, (p-79)(p^2 + 7pq + 49), is the closest to the correct factorization, with the only error being the first term, which should be (p-7) instead of (p-79).

Therefore, the provided options do not include the correct factorization, and the polynomial is reducible, not irreducible as suggested by Option D.