\( \left. \begin{array} { l } { 1 + a ^ { 7 } } \\ { \frac { 1 + a ^ { 7 } } { 1 + a } = ( 1 ) ^ { 6 } - ( 1 ) ^ { 5 } a + ( 1 ) ^ { 4 } a ^ { 2 } - ( - 1 ) ^ { 3 } a ^ { 3 } + ( 1 ) ^ { 2 } a ^ { 4 } + ( 1 ) a ^ { 5 } + c } \\ { = 1 - a + a ^ { 2 } - a ^ { 3 } + a ^ { 4 } - a ^ { 5 } + a ^ { 6 } } \\ { 1 + a ^ { 7 } = ( 1 + a ) ( 1 - a + a ^ { 2 } - a ^ { 3 } + a ^ { 4 } - a ^ { 5 } + a ^ { 6 } ) } \end{array} \right. \)

Here’s a fun and curious tidbit about polynomials: they’ve been around for centuries! Ancient civilizations, like the Babylonians and Greeks, utilized early forms of polynomial equations for practical purposes such as land measurements and astronomy. Fast forward to the Renaissance, when mathematicians like Tartaglia and Cardano laid the groundwork for polynomial theory, leading to the rich field of algebra we enjoy today. Now, let’s talk about those pesky common mistakes when working with polynomial factorization: people often forget to check for the degree of the polynomial correctly. When dealing with expressions like \(1 + a^7\), it’s easy to overlook whether your factors are indeed correct, especially when simplifying. Double-checking each step and ensuring all terms are accounted for will save you from confusion—it’s like a treasure hunt for math enthusiasts!