Ask AI: n⁴-100 factor

To factor the expression \( n^4 - 100 \), we can start by recognizing that \( 100 \) is a perfect square, which we can write as \( 10^2 \). The expression then becomes: \[ n^4 - 10^2 \] This fits the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = n^2 \) and \( b = 10 \): \[ n^4 - 10^2 = (n^2 - 10)(n^2 + 10) \] At this point, we need to check if the factors can be further simplified. 1. **Factor \( n^2 - 10 \):** \( n^2 - 10 \) is already simplified, as \( 10 \) is not a perfect square, and it cannot be factored further over the real numbers. 2. **Factor \( n^2 + 10 \):** Similarly, \( n^2 + 10 \) is already simplified, as \( 10 \) is not a perfect square, and it cannot be factored further over the real numbers. Therefore, the completely factored form of \( n^4 - 100 \) over the real numbers is: \[ (n^2 - 10)(n^2 + 10) \] If you are looking for a factorization over the complex numbers, you could factor further using imaginary numbers: \[ n^2 - 10 = n^2 - (\sqrt{10})^2 = (n - \sqrt{10})(n + \sqrt{10}) \] \[ n^2 + 10 = n^2 + (\sqrt{10}i)^2 = (n - \sqrt{10}i)(n + \sqrt{10}i) \] However, over the real numbers, the primary factorization remains: \[ (n^2 - 10)(n^2 + 10) \]