(2) \( 3 x^{4}-3 x^{2}-27 x^{2}+27 \) \( =\left(3 x^{4}+27\right)\left(-3 x^{2}-27 x^{2}\right)=\left(3 x^{4}\right. \) \( =3\left(x^{4}+9\right)-3 x^{2}(1+9)=3 x^{2}(6 \) \( = \)

To simplify the expression \( 3x^4 - 30x^2 + 27 \), you can start by factoring out the common factors. The expression can be rewritten as \( 3(x^4 - 10x^2 + 9) \). Next, notice that \( x^4 - 10x^2 + 9 \) can be treated as a quadratic in terms of \( y = x^2 \), leading to the factoring of \( (y - 1)(y - 9) \)—or in terms of \( x^2 \), \( (x^2 - 1)(x^2 - 9) \). Now, you can break this down further into two more factors: \( (x - 1)(x + 1)(x - 3)(x + 3) \). So, ultimately your simplified expression is: \[ 3(x-1)(x+1)(x-3)(x+3) \]. Perfect for finding roots or analyzing the graph! For fun, did you know that factoring polynomials like this has been around for centuries? Ancient Babylonian mathematicians were solving quadratic equations well before the time of algebra as we know it today! They developed early techniques for manipulations similar to factoring, demonstrating a keen understanding of numbers long before modern symbol notation. And here's a practical tip: when you have a polynomial to factor, always look for a greatest common factor first—this can simplify your work significantly! Also, common mistakes include misidentifying terms when rearranging or failing to account for signs, which can lead to incorrect factoring. Always double-check your work by expanding the factored form!