Chapter 5, Problem 17RE

Let $a_1,a_2,a_3,a_4$ be the first four terms of an arithmetic sequence. Show that $a_1{a}_4+3a_2{a}_3=2\left(a_1{a}_3+a_2{a}_4\right)$ and use this fact to factor the polynomial $f\left(x\right)=3x^2-2\left(a_1{a}_3+a_2{a}_4\right)x+a_1{a}_2{a}_3{a}_4$. [Students of calculus should recognize $f\left(x\right)$ as the derivative of $g(x)=x^3-\left(a_1{a}_3+a_2{a}_4\right)x^2+a_1{a}_2{a}_3{a}_{4}x=x\left(x-a_1{a}_3\right)\left(x-a_2{a}_4\right)$. Finding classes of polynomials that, together with their derivative and second derivative, factor over the rationals is a very hard problem. ]