## Sgstudent 3 years ago Given that 5x^3 + 6x^2 -x + 2 = (x+1) (x-2) Q(x) + ax + b find the value of a and b. Question on Identities :)

1. Ishaan94

Q(x) is an unknown polynomial?

2. Sgstudent

yea

3. nphuongsun93

then... a = -1, b = 2?...dont think it should be this simple

4. Sgstudent

5. Sgstudent

answer is a = 20 b = 24

6. Sgstudent

but i cant get it :(

7. Ishaan94

if Q(x) = 5x-9 ?

8. Sgstudent

o..o maybe you could work out the full solution and see if it matches my answer :P

9. Sgstudent

Thank you.

10. Ishaan94

\[ (x+​1)*​(x+​2)*​(5*​x-​9) = 5x^3 + 6x^2 -17x - 18 + 16x +20=5x^3 + 6x^2 -x + 2\]

11. Ishaan94

@Sgstudent

12. Sgstudent

oo then how should i carry on

13. Ishaan94

hmm a is 16 and b is 20 for me

14. Sgstudent

check for any careless mistake? :O

15. Ishaan94

no i don't think so

16. Sgstudent

17. Ishaan94

i didn't do it on my notebook, in head.

18. Ishaan94

\[(x+​1)(x+​2)(5x−​9) +ax + b = 5x^3+6x^2−17x−18+ ax+b\]

19. cwrw238

i did it also and got same result as ishaan

20. Sgstudent

i need the working :(

21. Sgstudent

or rather simple explanation towards the steps

22. cwrw238

equating coefficients of x gives -1 = 10 - 27 + a a = 27-11 = 16

23. Ishaan94

use q(x) as px +q, suggested by you earlier if you really need to show working

24. Sgstudent

can i let Q(x) = px + q will i be able to get the answer?

25. Sgstudent

oh but i cant seem to get the answer somehow :(

26. Sgstudent

i get decimal numbers :(

27. Ishaan94

show me your working i will try to guide you

28. Sgstudent

kk i try again

29. Ishaan94

well you can get multiple answers, maybe.

30. Sgstudent

only a and b

31. Sgstudent

(x+1) (x-2) *********

32. Sgstudent

Ishaan i edited the question

33. Sgstudent

34. cwrw238

i can see how ishaan got p = 5 because of the 5x^3 but the q = -9 is not obvious = 5x^3 + 15x^2 + 10x + qx^2 + 3qx + 2q + ax + b = 5x^3 + (15 + q)x^2 + (10 + 3q + a)x + 2q + b compare coefficients with 5x^3 + 6x^2 - x + 2 of x^2: q + 15 = 6 so q = -9 of x: 10 + a + 3q = 10 + a - 27 = a - 17 and a -17 = -1 so a = 16 of x^0 : 2q + b = 2 so -18 + b = 2 so b = 20 a=16, b = 20

35. Sgstudent

Thanks :)

36. cwrw238

yw