基礎数学 No.3 2005. 4.22
1.3 式の展開と因数分解（解答） 担当：市原
問題 7 展開公式を利用して,次の値を求めなさい. (1) 7×1997 = 7×(2000−3) = 14000−21 = 13979 (2) 1022 = (100 + 2)2 = 10000 + 400 + 4 = 10404 (3) 392= (40−1)2 = 1600−80 + 1 = 1521
(4) 103×97 = (100 + 3)(100−3) = 10000−9 = 9991
(5) 104×99 = (100 + 4)(100−1) = 10000 + 300−4 = 10296 (6) 1023 = (100 + 2)3 = 1000000 + 60000 + 1200 + 8 = 1061208 (7) 993= (100−1)3 = 1000000−30000 + 300−1 = 970299
問題 8 次の各式を因数分解しなさい．
(1)x2−3x−40 = (x−8)(x+ 5)
(2)z8−5z4+ 4 = (z4−4)(z4−1)
= (z2+ 2)(z2−2)(z2+ 1)(z2−1) = (z2+ 2)(z2−2)(z2+ 1)(z+ 1)(z−1) (3) 6x2+ 5x+ 1 = (3x+ 1)(2x+ 1)
(4) 8x3+ 27 = (2x)3+ 33= (2x+ 3)((2x)2−6x+ 9) = (2x+ 3)(4x2−6x+ 9) (5) 6t2+ 10t+ 4 = 2(3t2+ 5t+ 2) = 2(3t+ 2)(t+ 1)
(6) 3A3−7A2+ 2A=A(3A2−7A+ 2) =A(3A−1)(A−2)
問題 9 次のxに関する式をa(x−b)2+cの形に表しなさい．
(1) 3(x2+ 6x+ 9)−1 = 3(x+ 3)2−1 (2)x2−2x+ 3 =¡
x2−2x+ 1¢
+ 2 = (x−1)2+ 2 (3) 4x2−8x−1 = 4¡
x2−4x¢
−1 = 4¡
x2−4x+ 4¢
−16−1 = 4 (x−2)2−17
