1234561 2 3 4 5 6赋值法求二项式系数和例 1.若(1-2 x) 6 =a +a x +a x 2 +a x 3 +a x 4 +a x 5 +a x0 1 2 3 4 5 66，则（1）a +a +a +a +a +a +a = 0 1 2 3 4 5 6_；（2）a +a +a +a = 0 2 4 6_；（3）a +a +a = 1 3 5_；（4）| a | +| a | +| a | +| a | +| a | +| a | +| a |= 0 1 2 3 4 5 6_；（5）( a +a +a +a ) 0 2 4 62-(a +a +a ) 1 3 52=_；（6）a +2 a +2 0 12a +223a +234a +245a +256a =6_；（7）a -2 a +2 2 a -23 a +2 4 a -25 a +2 6 a = 0 1 2 3 4 5 6_；（8）（9）a +2 a +3a +4 a +5a +6 a = 1 2 3 4 5 6a -2 a +3a -4 a +5a -6 a = 1 2 3 4 5 6_；_；（10）a +01 1 1 1 1 1 a + a + a + a + a + a =2 3 4 5 6 7_；（11）-a +01 1 1 1 1 1 a - a + a - a + a - a =2 3 4 5 6 7_.解析：（1）令x =1，则a +a +a +a +a +a +a =1 0 1 2 3 4 5 6；（2）令x =-1，则 a -a +a -a +a -a +a =360 1 2 3 4 5 6=729，故a +a +a +a = 0 2 4 61 +7291=365；（3）由（1）（2）知a +a +a = 1 3 51 -7291=-364；234551234560123456123 4 5 61 2 3 4 5 6（4）| a | +| a | +| a | +| a | +| a | +| a | +| a |=a -a +a -a +a -a +a =729 0 1 2 3 4 5 6 0 1 2 3 4 5 6；（5）平方差公式，( a +a +a +a ) 2 -(a +a +a ) 2 =( a +a +a +a +a +a +a )(a -a +a -a +a -a +a ) 0 2 4 6 1 3 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6=1 36 =729（6）令x =2，则a +2a +2 0 12a +232a +234a +254a +256a =(1-4) 66=729；（7）令x =-2，则a -2a +2 2 a -23 a +2 4 a -25 a +26 a =(1+4) 6 =56 =15625 0 1 2 3 4 5 6；（8）对原式求导，a +2a x +3a x 1 2 32+4a x43+5a x54+6a x65=6(1-2 x)5( -2) ，令 x =1 ，则a +2a +3a +4a +5a +6a =6(1-2) 1 2 3 4 5 65( -2) =12；（9）对原式求导，a +2 a x +3a x +4a x +5a x +6a x =6(1-2 x ) (-2)1 2 3 4 5 6a -2a +3a -4a +5a -6a =6(1+2) 5 ( -2) =-2916则；1 2 3 4 5 6，令x =-1，（ 10 ）原式可由a x +01 1 1 1 1 1 1a x 2 + a x 3 + a x 4 + a x 5 + a x 6 + a x 7 =- (1-2 x) 2 3 4 5 6 7 147求导得到，令x =1，则1 1 1 1 1 1 1 1 a + a + a + a + a + a + a =- (1-2) 7 =2 3 4 5 6 7 14 14；（ 11 ）原式可由a x +01 1 1 1 1 1 1a x2 + a x3 + a x4 + a x5 + a x6 + a x7 =- (1 -2 )x7 2 3 4 5 6 7 14求导得到，令x =-1，则-a +01 1 1 1 1 1 1 3 a - a + a - a + a - a =- + (1 =2-)7 2 3 4 5 6 7 14 147=-218714.