Expert: Paul Klarreich - 2/28/2008

Question
The remainder of the division of a polynomial A(x) by ( x+1) is 3 and that by (x-3) is 23.
find the remainder of A(x) by ( x^2-2x-3)

Answer
Questioner:   metalx
Category:  Advanced Math
Private:  No
 
Subject:  polynomial remainder problem
Question:  The remainder of the division of a polynomial A(x) by ( x+1) is 3 and that by (x-3) is 23.
find the remainder of A(x) by (x^2-2x-3)
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Hi, Metty, [Is that what your mother likes to call you?]

Use these facts:

The Remainder Theorem:

If the remainder of the division of  A(x) by ( x+1) is 3, then A(-1) = 3
If the remainder of the division of  A(x) by (x-3) is 23, then A(3) = 23.

x^2 - 2x - 3 = (x +1)(x - 3), by coincidence.

Now when A(x) is divided by x^2 - 2x - 3, the remainder is a linear function, not necessarily a constant.  So write:

A(x) = Q(x)(x + 1)(x - 3) + R1x + R0
Now put in  -1, 3:

A(-1) = Q(-1)(-1 + 1)(-1 - 3) + R1(-1) + R0 = 3
A(-1) =         0             + R1(-1) + R0 = 3
...............
A(3)  = Q(3)(3 + 1)(3 - 3)    + R1(3) + R0 = 23
A(3)  =         0             + R1(3) + R0 = 23

You have two equations now, for R1 and R2:


- 1R1 + R0 = 3
 3R1 + R0 = 23

Solve those and get R1 = 5,  R0 = 8

Your remainder is  5x + 8