polynomial problem

[fiery123]

When a polynomial P(x) is divided by x-2, the remainder is 3. When it is divided by x-3, the remainder is 2. Find the remainder when P(x) is divided by (x-2)(x-3).

[Rassis]

If I understood it right, the answer is:

Given: P(x)/(x-2) = 3 and P(x)/(x-3) = 2
One gets: P(x)/[(x-2).(x-3)] = 1/[(x-2).(x-3)]/P(x) = 1/[(x-2)/P(x).(x-3)/P(x)] = 1/[(1/3).(1/2)] = 6

[fiery123]

sry i just found a mistake in my first post. I edited it already. The answer is -x+5 according to my book. But i have no idea how to work it out.

[Glaysher]

If I understood it right, the answer is:

Given: P(x)/(x-2) = 3 and P(x)/(x-3) = 2
One gets: P(x)/[(x-2).(x-3)] = 1/[(x-2).(x-3)]/P(x) = 1/[(x-2)/P(x).(x-3)/P(x)] = 1/[(1/3).(1/2)] = 6

No he's talking about the remainder on division. They don't divide exactly

eg 21/5 = 4 remainder 1

[Glaysher]

sry i just found a mistake in my first post. I edited it already. The answer is -x+5 according to my book. But i have no idea how to work it out.

I suspect it has something to do with the remainder theorem but I haven't worked it out yet

[Rassis]

Thanks Glaysher for the correction.

[krtxmrtz]

You have:

P(x) / (x - 2) = R(x) + 3
P(x) / (x - 3) = S(x) + 2

where R(x) and S(x) are the quotient polynomials. Therefore,

P(x) / [(x - 2)(x - 3)] = -P(x) / (x - 2) + P(x) / (x - 3) = -[R(x) + 3] + [S(x) + 2] =

...(call T = S - R)...

= T(x) - 1

So the remainder you want is -1

[krtxmrtz]

You have:

P(x) / (x - 2) = R(x) + 3
P(x) / (x - 3) = S(x) + 2

where R(x) and S(x) are the quotient polynomials.

Hi there folks. The above statement is a mistake on my side, I didn't really slow down to think what I was typing. The correct statements are:

P(x) = R(x)*(x - 2) + 3
P(x) = S(x)*(x - 3) +2

and this, of course, invalidates all I had previously posted.

If I find the solution I'll come back, but at the moment it's pretty late here and I must rush off to bed...
Sorry about it, I hope I haven't misled anyone, but the error is obvious if only you read that with care.

[Glaysher]

P(x) / [(x - 2)(x - 3)] = -P(x) / (x - 2) + P(x) / (x - 3)

= -[R(x) + 3/(x - 2)] + [S(x) + 2/(x - 3)]

= [S(x) - R(x)] + 2/(x - 3) - 3/(x - 2)

= [S(x) - R(x)] + [2(x - 2)]/[(x -2)(x - 3)] - [3(x -3)]/[(x - 2)(x - 3)]

= [S(x) - R(x)] + [-x + 5]/[(x - 2)(x - 3)]

So remaninder is -x + 5

as P(x) = [S(x) - R(x)][(x - 2)(x - 3)] + (-x + 5)

[fiery123]

hmm. ok i still don't get it. But it's probably too hard for me anyway. So thanks for all your help.

[Glaysher]

Working was skipped out in krtxmrtz first flawed answer. He used partial fractions to show that 1/[(x - 2)(x - 3)] = -1/(x - 2) + 1/(x - 3)

Then multiplied both sides by P(x). I used this in my corrected version.

Was there any other parts of the solution you would like further explanations for?