1.因式分解例1已知a是正整数,如果关于x的方程x~3+(a+17)x~2+(38-a)x-56=0的三个根都是整数,求a的值及方程的整数根.(2007年全国初中数学联赛)解将方程的左边分解因式,得(x-1)[x~2+(a+18)x+56]=0.因为a是正整数,所以关于x的方程x~2+(a+18)x+56=0的判别式Δ=(a+18)~2-224>0,它一定有两个不同的实数根.而原方程的根都是整数, 1. Factorization Example 1 It is known that a is a positive integer. If all three roots of the equation x ~ 3 + (a + 17) x ~ 2 + (38-a) x-56 = 0 for x are integers, find (a + 18) x + 56] = 0. Since the left side of the equation is decomposed into a factor of (x-1) [x 2 + (a + 18) x + 56] Since a is a positive integer, the discriminant Δ = (a + 18) ~ 2-224> 0 for the equation x ~ 2 + (a + 18) x + 56 = 0 for x must have two distinct real roots. The roots of the original equation are integers,