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).
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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).
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
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.
No he's talking about the remainder on division. They don't divide exactlyQuote:
Originally Posted by Rassis
eg 21/5 = 4 remainder 1
I suspect it has something to do with the remainder theorem but I haven't worked it out yetQuote:
Originally Posted by fiery123
Thanks Glaysher for the correction.
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
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:Quote:
Originally Posted by krtxmrtz
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... :wave:
Sorry about it, I hope I haven't misled anyone, but the error is obvious if only you read that with care.
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)
hmm. ok i still don't get it. But it's probably too hard for me anyway. So thanks for all your help.
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?