## Callisto 3 years ago Polynomials #1 Question 41 It is given that f(3x) = 54x^3 -27x^2 +px +q. When f(x) is divided by (x-3), the remainder is 42. Find the remainder when f(x/3) is divided by x-9 *Note: I'm helping my sister but I'm in trouble too :|*

1. FoolForMath

I thought someone was typing an answer.

2. cwtan

TT bluring....... btw i found -4.... I duno whether it is right or wrong...

3. Callisto

The answer is not -4 :|

4. cwtan

younger sister? it's complicated... LOL

5. Callisto

Younger sister...

6. Ackhat

p+q=6?

7. Callisto

Nope :| p+q=15

8. cwtan

Big trouble

9. Ackhat

oh yes 15

10. Ackhat

i know wait

11. Ackhat

:) why my answer is 42

12. Callisto

It is!!!!

13. Ackhat

ohhohohoh

14. Ishaan94

Lol this question is funny.

15. Callisto

How did you get that answer? @Ackhat

16. ash2326

We have $f(3x)=54x^3-27x^2+px+q$ It can be written as $f(3x)=2(3x)^3-3(3x)^2+\frac p 3 (3x) +q$ so $f(x)=2x^3-3x^2+\frac p 3 x+q$ We are given that when f(x) is divided by (x-3) the remainder is 42 so Using remainder theorem $f(3)=42$ Now you can find P+q from here, Next find $$f(\frac x 3)$$ to find the remainder when f(x/3) is divided by (x-9) put x=9 in f(x/3)

17. Ackhat

f(x)=2x^3-3x^2+p/3x+q

18. Callisto

.... I've thought about it.....I swear......

19. Ishaan94

f(x)/(x-3)=k+42/(x-3) 3f(x/3)/(x-9)= k + 3*42/(x-9)

20. Ishaan94

:D

21. Ackhat

i simpy divide by x-3 and got p+q=15 and after by x-9 and got that the reminder is p+q+27

22. Callisto

Hold on... @Ishaan94 How does it work: 3f(x/3)/(x-9)= k + 3*42/(x-9)? @Ackhat Do you mean you divided the equation f(x) by (x-3) first, then divide f(x) by (x-9)? or ...?

23. Ackhat

f(x)/(x-3) f(x/3)/(x-9)

24. Ishaan94

$\frac{f(x)}{x-3} = P(x) + \frac{42}{x-3}$$x = \frac{x}{3}$ $\frac{3\cdot f\left(x/3\right)}{x-9} = P(x) + \frac{42\cdot 3}{x-9}$ I love my solution <3

25. Callisto

Okay, got it. My calculation mistake :| @Ackhat

26. Callisto

@Ishaan94 More explanation is appreciated :) (sorry... I'm stupid :| )

27. FoolForMath

Beautiful solution Ishaan.

28. FoolForMath

Calli: Division Algorithm

29. Ishaan94

I am not a good teacher, Sorry. What part you didn't understand callisto?

30. Ishaan94

Thank you very much foolformath

31. Callisto

$\frac{3\cdot f\left(x/3\right)}{x-9} = P(x) + \frac{42\cdot 3}{x-9}$ ^ don't know where it comes..

32. Ackhat

Bezu

33. Ackhat

:)

34. cwtan

What a simple solution!!! Great job @Ishaan94 !!!!!

35. FoolForMath

Substitute $$x=\frac x3$$ in $\frac{f(x)}{x-3} = P(x) + \frac{42}{x-3}$

36. Callisto

Oh... Got it!!!! Thanks!!! Lovely solution :)

37. Ishaan94

Okay. $\large\frac{f\left(\frac{x}{3}\right)}{\frac{x}3-3} = P + \frac{42}{\frac x3-3}$Where P is any quadratic polynomial. $\large \implies \frac{f\left(\frac x3\right)}{\frac{x-9}3} = P + \frac{42}{\frac {x -9}3}$

38. FoolForMath

No, it is beautiful :|

39. Ishaan94

What's 'Bezu'? @Akchat

40. Ishaan94

@Ackhat *

41. Ackhat

never mind

42. Ackhat

ok this is a theorem

43. Callisto

The most wonderful thing is that my sister understands it :) Once again, thank you everyone :)