About Lesson
1. Calculate the remainder when
is divided by
.
试计算
除以
所得余数。


试计算


2. Given that
and
have the same remainder when divided by
, find the possible values of a.
已知
和
除以
所得余数相等,求a 的可能值。



已知



3. The expression
leaves a remainder of
and
when divided by
and
respectively. Calculate the value of a and b.
当式子
除以
和
所得余数分别为
和
. 试计算a 和 b 的值。
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]f (–2) = 19, f (1) = –8, 解联立[/bg_collapse]





当式子





[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]f (–2) = 19, f (1) = –8, 解联立[/bg_collapse]
4. The expression
leaves a remainder of k when divided by
and a remainder of
when divided by
. Show that
.
式子
当除以
时所得余数为 k 和除以
时所得余数为
. 试证
.
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]f (–2) = k, f (3) = 3k + 5,用代入消元法消掉k。[/bg_collapse]





式子





[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]f (–2) = k, f (3) = 3k + 5,用代入消元法消掉k。[/bg_collapse]
5. A polynomial
has a remainder 15 when divided by
, and 9 when divided by
. Find the remainder when
is divided by
.
一多项式
当除以
时所得余数为15, 及当除以
时所得余数为 9. 求当
除以
时所得余式.
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设
, 则
,
后解联立得a, b。余式为
[/bg_collapse]





一多项式





[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设




6. If deg
, and f (x) is divided by
,
,
, the remainders are
,
and
respectively. Find
.
若
为四次多项式,且当
除以
,
,
, 所得余数分别为
,
和
. 求
.
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设
, 则
,
后解联立得
,
. 把
,
代入
[/bg_collapse]








若









[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设








7.
where
are constants. If satisfy the following conditions:
其中
为常数。若其满足以下各条件:




(i) when divided by
, obtain the remainder
,
当 除以
, 所得余式为
,
(ii) when divided by
, we obtain the remainder
.
当 除以
, 所得余数为
.
Find the values of ,
and
.
求,
和
的值.
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]从 (i) 可得 . 从 (ii) 可得
. 代入
可得
,将
展开可得
和
的值[/bg_collapse]
8. When the polynomial
is divided by
,
, the remainder is
and
respectively. Find the remainder when
is divided by
.
当多项式
除以
,
, 所得余式分别为
和
. 求当
除以
所得余式.
[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设
,
可得
。余式为
[/bg_collapse]







当多项式







[bg_collapse view=”link” color=”#22A609″ expand_text=”hint” collapse_text=”close hint” ]设




Answer 答案:
(1) 13
(2) 1, 2
(3) 1, –5
(5) 2x + 13
(6)
(7) 1, 1, –7
(8)