simple polynomial question (1 Viewer)

[babygoose!]

hey guys! i have my half yearly on monday and i just cant seem to get this question out...its really simple but i'm stuck so any help would be good...thanx a lot!

when a certain polynomial P(x) is divided by (x + 1) and (x - 3) the respective remainders are 6 and -2. Find the remainder when this polynomial is divided by
x^2 - 2x - 3.

Please help me....!

 

[George W. Bush]

From vague recollection, I believe the answer is 6 + -2 = 4, but it's been a while so wait for a more versed mathematician to confirm/deny.

 

[CM_Tutor]

We know when a polynomial is divided by another polynomial of degfree n, the degree of the remainder will be at most n - 1. So, we can express P(x) as:

P(x) = (x² - 2x - 3)Q(x) + Ax + B, where Ax + B is the required remainder (A, B are real constants).

We know from the information given and the remainder theorem that P(-1) = 6 and P(3) = -2. Using these, we get:

P(-1) = [(-1)² - 2(-1) - 3]Q(-1) + A(-1) + B = 6
(1 + 2 - 3)Q(-1) - A + B = 6
B - A = 6 __________ (1)

P(3) = [(3)² - 2(3) - 3]Q(3) + A(3) + B = -2
(9 - 6 - 3)Q(3) + 3A + B = -2
3A + B = -2 __________ (2)

(1) - (2): -4A = 8
A = -2

Put A = -2 into (2): B = -2 - 3(-2) = 4

So, the required remainder is -2x + 4

 

[babygoose!]

wow! thanx so much..i remember how to do that now...!