Factor out the greatest common factor from the following polynomial. \( 7 x^{6}-63 x^{5}+35 x^{4} \)

To factor out the greatest common factor from the polynomial \( 7 x^{6}-63 x^{5}+35 x^{4} \), we first identify the GCF of the coefficients (7, -63, and 35), which is 7. Next, we look for the lowest power of \( x \) present in all terms, which is \( x^4 \). Now, we factor out \( 7 x^{4} \) from each term: \[ 7 x^{6} - 63 x^{5} + 35 x^{4} = 7 x^{4}(x^{2} - 9x + 5) \] So, the factored form of the polynomial is \( 7 x^{4}(x^{2} - 9x + 5) \). Next, let's break down some fascinating insights! Did you know that the development of polynomial factorization dates back to ancient civilizations? The Babylonians used geometric methods to tackle area problems that essentially involved polynomial operations, and their work laid the foundation for algebra today. Talk about roots of knowledge! When you're tackling polynomial problems, always look for the GCF first! It’s a nifty trick to simplify your work before diving into other methods like grouping or using the quadratic formula. A common mistake is forgetting to check for a GCF, which can lead to more complicated calculations. Keep your eyes peeled for simplicity!