What is the numerator in the sum? \( 2(x+1)(x-2)(x-4)+3(x+1)(x-2)(x-4) \) \( 2(x-2)(x-4)+3(x+1)(x-4) \) \( 2(x-2)+3(x+1) \)

To simplify the expression and find the numerator in the sum, first, notice that you can factor out the common terms in the initial expression. You can combine the first two terms: \[ (2 + 3)(x + 1)(x - 2)(x - 4) = 5(x + 1)(x - 2)(x - 4) \] From there, simplify it further and focus on the simplified structure. When you distribute, you'll end up with a polynomial whose numerator comprises the combined coefficients of each term. This builds precisely how we can express the polynomial components leading to the final sum. Ultimately, the numerator is derived from the combination of these factors in the polynomial produced from multiplication, leading to a useful polynomial form when fully expanded. Fun fact: this can often lead to surprising patterns when you explore the roots of the polynomial created through such sums. They often align nicely with the arithmetic sequences or exhibit symmetry based on the factors chosen!