## shubhamsrg 3 years ago Let p(x) be a polynomial such that when divided by (x−1) it leaves the remainder 2 and when divided by (x − 2) it leaves the remainder 1. What is the remainder when it is divided by (x − 1)(x − 2)?

1. mukushla

when we divide by a second degree polynomial the remainder will generally be linear.so the division statement becomes\[P(x)=(x-1)(x-2)q(x)+ax+b\]does that help?

2. mukushla

u just need to evaluate \(a\) and \(b\) regarding given conditions for problem

3. amistre64

using the remainder thrm P(1) = 2 ; and P(2) = 1 P(x) = -x+3 is my idea

4. shubhamsrg

@mukushla can we write like that ? a and b do not change in both eqns ?

5. mukushla

what do u mean by both equations?

6. shubhamsrg

using p(1) and p(2),we get 2 eqns.. well is it the ggeneralised form? how?

7. sauravshakya

I think that will give remainder in terms of x

8. shubhamsrg

hmm,,remainder shouldn't be some numerical ?

9. mukushla

when we divide by a linear polynomial the remainder will be a number. when we divide by a second degree polynomial the remainder will be a linear poly. and ...

10. shubhamsrg

ahh i see...yep..got it..

11. shubhamsrg

so we have p(1) = 2 = a + b and p(2) = 1 = 2a + b is that right ?

12. mukushla

yes thats right.

13. shubhamsrg

so rem is -x + 3 ! ahh..thank you sir..

14. mukushla

np :)