關於remainder theorem

The remainder theorem states that if a polynomial f(x) is divided by a linear function x - k, the remainder is f(k).
Proof:
In any division,
dividend = divisor X quotient + remainder
Let Q(x) be the quotient and R be remainder.
f(x) = (x - k) Q(x) + R
f(k) = (k - k) Q(x) + R
= 0 + R
= R
Note The degree of the remainder is always one less than the degree of the divisor.

圖片參考：http://library.thinkquest.org/C0110248/images/blueline.gif

Example 1

Find the remainder when x3 + 4x2 - 7x + 6 is divided by x - 1.


圖片參考：http://library.thinkquest.org/C0110248/images/iconans.gif
Let f(x) = x3 + 4x2 - 7x + 6
f(1) = 13 + 4 (1)2 - 7 + 6
= 4

圖片參考：http://library.thinkquest.org/C0110248/images/blueline.gif


Example 2

Given that the expression 2x3 + 3px2 - 4x + p has a remainder of 5 when divided by x + 2, find the value of p.


圖片參考：http://library.thinkquest.org/C0110248/images/iconans.gif
Let f(x) = 2x3 + 3px2 - 4x + p
f (-2) = 2(-2)3 + 3(-2)2p - 4(-2) + p = 5
13p - 8 = 5
13p = 13
p = 1


圖片參考：http://library.thinkquest.org/C0110248/images/blueline.gif

Example 3

If the expression ax4 + bx3 - x2 + 2x + 3 has remainder 4x + 3 when divided by x2 + x - 2, find the value of a and b.


圖片參考：http://library.thinkquest.org/C0110248/images/iconans.gif
Let f(x) = ax4 + bx3 - x2 + 2x + 3
x2 + x - 2 = (x + 2)(x - 1)

f(-2) = a(-2)4 + b(-2)3 - (-2)2 + 2(-2) + 3 = 4(-2) + 3
16a - 8b - 4 - 4 + 3 = -5
2a - b = 0 --------(1)

f(1) = a + b - 1 + 2 + 3 = 4(1) + 3
a + b = 3 --------(2)

(1) + (2) : 3a = 3
a = 1

when a = 1, b = 2.