笔算实在是不知道怎么算,就用MATLAB计算.我已经计算了当k=-10000到10000的每个值时能不能因式分解的情况了,发现只有4个k值能行,分别是k=-2或k=-1或k=0或k=1,因此怀疑k值只能有四种情况.但是你题目说k是正整数,所以只能是k=1.以下是当k=-100到100的情况(全部写进来装不下).
>> syms x ; k=-100:100; Z=x.^4 - x.^3 + k.*x.^2 - 2.*k.*x - 2;Z=factor(Z)
Z =
[ x^4 - x^3 - 100*x^2 + 200*x - 2, x^4 - x^3 - 99*x^2 + 198*x - 2, x^4 - x^3 - 98*x^2 + 196*x - 2, x^4 - x^3 - 97*x^2 + 194*x - 2, x^4 - x^3 - 96*x^2 + 192*x - 2, x^4 - x^3 - 95*x^2 + 190*x - 2, x^4 - x^3 - 94*x^2 + 188*x - 2, x^4 - x^3 - 93*x^2 + 186*x - 2, x^4 - x^3 - 92*x^2 + 184*x - 2, x^4 - x^3 - 91*x^2 + 182*x - 2, x^4 - x^3 - 90*x^2 + 180*x - 2, x^4 - x^3 - 89*x^2 + 178*x - 2, x^4 - x^3 - 88*x^2 + 176*x - 2, x^4 - x^3 - 87*x^2 + 174*x - 2, x^4 - x^3 - 86*x^2 + 172*x - 2, x^4 - x^3 - 85*x^2 + 170*x - 2, x^4 - x^3 - 84*x^2 + 168*x - 2, x^4 - x^3 - 83*x^2 + 166*x - 2, x^4 - x^3 - 82*x^2 + 164*x - 2, x^4 - x^3 - 81*x^2 + 162*x - 2, x^4 - x^3 - 80*x^2 + 160*x - 2, x^4 - x^3 - 79*x^2 + 158*x - 2, x^4 - x^3 - 78*x^2 + 156*x - 2, x^4 - x^3 - 77*x^2 + 154*x - 2, x^4 - x^3 - 76*x^2 + 152*x - 2, x^4 - x^3 - 75*x^2 + 150*x - 2, x^4 - x^3 - 74*x^2 + 148*x - 2, x^4 - x^3 - 73*x^2 + 146*x - 2, x^4 - x^3 - 72*x^2 + 144*x - 2, x^4 - x^3 - 71*x^2 + 142*x - 2, x^4 - x^3 - 70*x^2 + 140*x - 2, x^4 - x^3 - 69*x^2 + 138*x - 2, x^4 - x^3 - 68*x^2 + 136*x - 2, x^4 - x^3 - 67*x^2 + 134*x - 2, x^4 - x^3 - 66*x^2 + 132*x - 2, x^4 - x^3 - 65*x^2 + 130*x - 2, x^4 - x^3 - 64*x^2 + 128*x - 2, x^4 - x^3 - 63*x^2 + 126*x - 2, x^4 - x^3 - 62*x^2 + 124*x - 2, x^4 - x^3 - 61*x^2 + 122*x - 2, x^4 - x^3 - 60*x^2 + 120*x - 2, x^4 - x^3 - 59*x^2 + 118*x - 2, x^4 - x^3 - 58*x^2 + 116*x - 2, x^4 - x^3 - 57*x^2 + 114*x - 2, x^4 - x^3 - 56*x^2 + 112*x - 2, x^4 - x^3 - 55*x^2 + 110*x - 2, x^4 - x^3 - 54*x^2 + 108*x - 2, x^4 - x^3 - 53*x^2 + 106*x - 2, x^4 - x^3 - 52*x^2 + 104*x - 2, x^4 - x^3 - 51*x^2 + 102*x - 2, x^4 - x^3 - 50*x^2 + 100*x - 2, x^4 - x^3 - 49*x^2 + 98*x - 2, x^4 - x^3 - 48*x^2 + 96*x - 2, x^4 - x^3 - 47*x^2 + 94*x - 2, x^4 - x^3 - 46*x^2 + 92*x - 2, x^4 - x^3 - 45*x^2 + 90*x - 2, x^4 - x^3 - 44*x^2 + 88*x - 2, x^4 - x^3 - 43*x^2 + 86*x - 2, x^4 - x^3 - 42*x^2 + 84*x - 2, x^4 - x^3 - 41*x^2 + 82*x - 2, x^4 - x^3 - 40*x^2 + 80*x - 2, x^4 - x^3 - 39*x^2 + 78*x - 2, x^4 - x^3 - 38*x^2 + 76*x - 2, x^4 - x^3 - 37*x^2 + 74*x - 2, x^4 - x^3 - 36*x^2 + 72*x - 2, x^4 - x^3 - 35*x^2 + 70*x - 2, x^4 - x^3 - 34*x^2 + 68*x - 2, x^4 - x^3 - 33*x^2 + 66*x - 2, x^4 - x^3 - 32*x^2 + 64*x - 2, x^4 - x^3 - 31*x^2 + 62*x - 2, x^4 - x^3 - 30*x^2 + 60*x - 2, x^4 - x^3 - 29*x^2 + 58*x - 2, x^4 - x^3 - 28*x^2 + 56*x - 2, x^4 - x^3 - 27*x^2 + 54*x - 2, x^4 - x^3 - 26*x^2 + 52*x - 2, x^4 - x^3 - 25*x^2 + 50*x - 2, x^4 - x^3 - 24*x^2 + 48*x - 2, x^4 - x^3 - 23*x^2 + 46*x - 2, x^4 - x^3 - 22*x^2 + 44*x - 2, x^4 - x^3 - 21*x^2 + 42*x - 2, x^4 - x^3 - 20*x^2 + 40*x - 2, x^4 - x^3 - 19*x^2 + 38*x - 2, x^4 - x^3 - 18*x^2 + 36*x - 2, x^4 - x^3 - 17*x^2 + 34*x - 2, x^4 - x^3 - 16*x^2 + 32*x - 2, x^4 - x^3 - 15*x^2 + 30*x - 2, x^4 - x^3 - 14*x^2 + 28*x - 2, x^4 - x^3 - 13*x^2 + 26*x - 2, x^4 - x^3 - 12*x^2 + 24*x - 2, x^4 - x^3 - 11*x^2 + 22*x - 2, x^4 - x^3 - 10*x^2 + 20*x - 2, x^4 - x^3 - 9*x^2 + 18*x - 2, x^4 - x^3 - 8*x^2 + 16*x - 2, x^4 - x^3 - 7*x^2 + 14*x - 2, x^4 - x^3 - 6*x^2 + 12*x - 2, x^4 - x^3 - 5*x^2 + 10*x - 2, x^4 - x^3 - 4*x^2 + 8*x - 2, x^4 - x^3 - 3*x^2 + 6*x - 2, (x - 1)*(x^3 - 2*x + 2), (x^2 - x + 1)*(x^2 - 2), (x + 1)*(x^3 - 2*x^2 + 2*x - 2), (x^2 - x - 1)*(x^2 + 2), x^4 - x^3 + 2*x^2 - 4*x - 2, x^4 - x^3 + 3*x^2 - 6*x - 2, x^4 - x^3 + 4*x^2 - 8*x - 2, x^4 - x^3 + 5*x^2 - 10*x - 2, x^4 - x^3 + 6*x^2 - 12*x - 2, x^4 - x^3 + 7*x^2 - 14*x - 2, x^4 - x^3 + 8*x^2 - 16*x - 2, x^4 - x^3 + 9*x^2 - 18*x - 2, x^4 - x^3 + 10*x^2 - 20*x - 2, x^4 - x^3 + 11*x^2 - 22*x - 2, x^4 - x^3 + 12*x^2 - 24*x - 2, x^4 - x^3 + 13*x^2 - 26*x - 2, x^4 - x^3 + 14*x^2 - 28*x - 2, x^4 - x^3 + 15*x^2 - 30*x - 2, x^4 - x^3 + 16*x^2 - 32*x - 2, x^4 - x^3 + 17*x^2 - 34*x - 2, x^4 - x^3 + 18*x^2 - 36*x - 2, x^4 - x^3 + 19*x^2 - 38*x - 2, x^4 - x^3 + 20*x^2 - 40*x - 2, x^4 - x^3 + 21*x^2 - 42*x - 2, x^4 - x^3 + 22*x^2 - 44*x - 2, x^4 - x^3 + 23*x^2 - 46*x - 2, x^4 - x^3 + 24*x^2 - 48*x - 2, x^4 - x^3 + 25*x^2 - 50*x - 2, x^4 - x^3 + 26*x^2 - 52*x - 2, x^4 - x^3 + 27*x^2 - 54*x - 2, x^4 - x^3 + 28*x^2 - 56*x - 2, x^4 - x^3 + 29*x^2 - 58*x - 2, x^4 - x^3 + 30*x^2 - 60*x - 2, x^4 - x^3 + 31*x^2 - 62*x - 2, x^4 - x^3 + 32*x^2 - 64*x - 2, x^4 - x^3 + 33*x^2 - 66*x - 2, x^4 - x^3 + 34*x^2 - 68*x - 2, x^4 - x^3 + 35*x^2 - 70*x - 2, x^4 - x^3 + 36*x^2 - 72*x - 2, x^4 - x^3 + 37*x^2 - 74*x - 2, x^4 - x^3 + 38*x^2 - 76*x - 2, x^4 - x^3 + 39*x^2 - 78*x - 2, x^4 - x^3 + 40*x^2 - 80*x - 2, x^4 - x^3 + 41*x^2 - 82*x - 2, x^4 - x^3 + 42*x^2 - 84*x - 2, x^4 - x^3 + 43*x^2 - 86*x - 2, x^4 - x^3 + 44*x^2 - 88*x - 2, x^4 - x^3 + 45*x^2 - 90*x - 2, x^4 - x^3 + 46*x^2 - 92*x - 2, x^4 - x^3 + 47*x^2 - 94*x - 2, x^4 - x^3 + 48*x^2 - 96*x - 2, x^4 - x^3 + 49*x^2 - 98*x - 2, x^4 - x^3 + 50*x^2 - 100*x - 2, x^4 - x^3 + 51*x^2 - 102*x - 2, x^4 - x^3 + 52*x^2 - 104*x - 2, x^4 - x^3 + 53*x^2 - 106*x - 2, x^4 - x^3 + 54*x^2 - 108*x - 2, x^4 - x^3 + 55*x^2 - 110*x - 2, x^4 - x^3 + 56*x^2 - 112*x - 2, x^4 - x^3 + 57*x^2 - 114*x - 2, x^4 - x^3 + 58*x^2 - 116*x - 2, x^4 - x^3 + 59*x^2 - 118*x - 2, x^4 - x^3 + 60*x^2 - 120*x - 2, x^4 - x^3 + 61*x^2 - 122*x - 2, x^4 - x^3 + 62*x^2 - 124*x - 2, x^4 - x^3 + 63*x^2 - 126*x - 2, x^4 - x^3 + 64*x^2 - 128*x - 2, x^4 - x^3 + 65*x^2 - 130*x - 2, x^4 - x^3 + 66*x^2 - 132*x - 2, x^4 - x^3 + 67*x^2 - 134*x - 2, x^4 - x^3 + 68*x^2 - 136*x - 2, x^4 - x^3 + 69*x^2 - 138*x - 2, x^4 - x^3 + 70*x^2 - 140*x - 2, x^4 - x^3 + 71*x^2 - 142*x - 2, x^4 - x^3 + 72*x^2 - 144*x - 2, x^4 - x^3 + 73*x^2 - 146*x - 2, x^4 - x^3 + 74*x^2 - 148*x - 2, x^4 - x^3 + 75*x^2 - 150*x - 2, x^4 - x^3 + 76*x^2 - 152*x - 2, x^4 - x^3 + 77*x^2 - 154*x - 2, x^4 - x^3 + 78*x^2 - 156*x - 2, x^4 - x^3 + 79*x^2 - 158*x - 2, x^4 - x^3 + 80*x^2 - 160*x - 2, x^4 - x^3 + 81*x^2 - 162*x - 2, x^4 - x^3 + 82*x^2 - 164*x - 2, x^4 - x^3 + 83*x^2 - 166*x - 2, x^4 - x^3 + 84*x^2 - 168*x - 2, x^4 - x^3 + 85*x^2 - 170*x - 2, x^4 - x^3 + 86*x^2 - 172*x - 2, x^4 - x^3 + 87*x^2 - 174*x - 2, x^4 - x^3 + 88*x^2 - 176*x - 2, x^4 - x^3 + 89*x^2 - 178*x - 2, x^4 - x^3 + 90*x^2 - 180*x - 2, x^4 - x^3 + 91*x^2 - 182*x - 2, x^4 - x^3 + 92*x^2 - 184*x - 2, x^4 - x^3 + 93*x^2 - 186*x - 2, x^4 - x^3 + 94*x^2 - 188*x - 2, x^4 - x^3 + 95*x^2 - 190*x - 2, x^4 - x^3 + 96*x^2 - 192*x - 2, x^4 - x^3 + 97*x^2 - 194*x - 2, x^4 - x^3 + 98*x^2 - 196*x - 2, x^4 - x^3 + 99*x^2 - 198*x - 2, x^4 - x^3 + 100*x^2 - 200*x - 2]
得到能分解因式的只有4个：
(x-1)(x^3-2x+2) = x^4-x^3-2x^2+4x-2
(x^2-x+1)(x^2-2) = x^4-x^3-x^2+2x-2
(x+1)(x^3-2x^2+2x-2) = x^4-x^3-2
(x^2-x-1)(x^2+2) = x^4-x^3+x^2-2x-2
即k=-2或k=-1或k=0或k=1.
因为题设说k是正整数,所以k=1.