Factor. Assume that variables used as exponents represent positive integers.
-$$x ^ { 4 n } - 1$$
A) $\left( x ^ { 2 n } + 1 \right) \left( x ^ { n } - 1 \right) ^ { 2 }$
B) $\left( x ^ { 2 n } + 1 \right) \left( x ^ { n } + 1 \right) \left( x ^ { n } - 1 \right)$
C) $\left( x ^ { 2 n } + 1 \right) \left( x ^ { 2 n } - 1 \right)$
D) $\left( x ^ { n } + 1 \right) ^ { 2 } \left( x ^ { n } - 1 \right) ^ { 2 }$

Question

Quizplus · Accepted Answer

The expression $x^{4n} - 1$ can be factored using the difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$, where $a = x^{2n}$ and $b = 1$. This gives us $(x^{2n} + 1)(x^{2n} - 1)$. The term $x^{2n} - 1$ can be further factored using the same formula, where $a = x^n$ and $b = 1$, resulting in $(x^n + 1)(x^n - 1)$. Therefore, the complete factorization is $(x^{2n} + 1)(x^n + 1)(x^n - 1)$, which matches choice B.

Factor Assume That Variables Used as Exponents Represent Positive Integers x4n−1x ^ { 4 n } - 1x4n−1

Factor Assume That Variables Used as Exponents Represent Positive Integers $x ^ { 4 n } - 1$