问题1.将x 3 + 3x 2 + 3x + 1除以除法求余数
(i)x + 1
Solution:
x + 1 = 0
x = −1
Therefore remainder will be f(x):
f(−1) = (−1)3 + 3(−1)2 + 3(−1) + 1
= −1 + 3 − 3 + 1
= 0
(ii)x – 1/2
Solution:
x – 1/2 = 0
x = 1/2
Therefore remainder will be f(x):
f(1/2) = (1/2)3 + 3(1/2)2 + 3(1/2) + 1
= (1/8) + (3/4) + (3/2) + 1
= 27/8
(iii)x
Solution:
x = 0
Therefore remainder will be f(x):
f(0) = (0)3 + 3(0)2 + 3(0) + 1
= 1
(iv)x +圆周率
Solution:
x + pi = 0
x = −pi
Therefore remainder will be f(x):
f(−pi) = (−pi)3 + 3(−pi)2 + 3(−pi) + 1
= −pi3 + 3pi2 − 3pi + 1
(v)5 + 2x
Solution:
5 + 2x = 0
2x = −5
x = -5/2
Therefore remainder will be f(x) :
f(-5/2) = (-5/2)3 + 3(-5/2)2 + 3(-5/2) + 1
= (-125/8) + (75/4) – (15/2) + 1
= -27/8
问题2.当x 3 − ax 2 + 6x − a除以x – a时,求出余数。
Solution:
Let f(x) = x3 − ax2 + 6x − a
x − a = 0
∴ x = a
Therefore remainder will be f(x):
f(a) = (a)3 − a(a2) + 6(a) − a
= a3 − a3 + 6a − a
= 5a
问题3.检查7 + 3x是否是3x 3 + 7x的因数。
Solution:
7 + 3x = 0
3x = −7
x = -7/3
Therefore remainder will be f(x):
f(-7/3) = 3(-7/3)3 + 7(-7/3)
= – (343/9) + (-49/3)
= (-343- (49) * 3)/9
= (-343 – 147)/9
= – 490/9 ≠ 0
∴ 7 + 3x is not a factor of 3x3 + 7x