Reminder theorem

2008-10-25 7:32 pm

a) write down the highest possible degree of the reminder when a polynomial P(x) is divided by (x-1)(x+2).

b) It is given that when P(x) is divided by x-1 and x+2, the reminders are -4 and -28 repectively. Find the reminder when P(x) is divided by(x-1)(x+2)

回答 (1)

Uncle Michael

2008-10-25 7:59 pm

✔ 最佳答案

a)
Highest possible degree of the reminder = 1

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b)
Let Q(x) and (ax + b) be the quotient and remainder respectively when P(x) is divided by (x - 1)(x + 2).
Then, P(x) = (x - 1)(x + 2)Q + ax + b

When P(x) is divided by x - 1, the reminder is -4.
P(1) = -4
(1 - 1)(1 + 2)Q(1) + a(1) + b = -4
a + b = -4 ...... (1)

When P(x) is divided by x + 2, the reminder is -28.
P(-2) = -28
(-2 - 1)(-2 + 2)Q(-2) + a(-2) + b = -4
-2a + b = -28 ...... (2)

(1) - (2):
3a = 24
a = 8

Put a = 6 into (1):
(8) + b = -4
b = -12

Ans: The required remainder = 8x - 12
=

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