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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 :|*

Mathematics
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I thought someone was typing an answer.
TT bluring....... btw i found -4.... I duno whether it is right or wrong...
The answer is not -4 :|

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Other answers:

younger sister? it's complicated... LOL
Younger sister...
p+q=6?
Nope :| p+q=15
Big trouble
oh yes 15
i know wait
:) why my answer is 42
It is!!!!
ohhohohoh
Lol this question is funny.
How did you get that answer? @Ackhat
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)
f(x)=2x^3-3x^2+p/3x+q
.... I've thought about it.....I swear......
f(x)/(x-3)=k+42/(x-3) 3f(x/3)/(x-9)= k + 3*42/(x-9)
:D
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
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 ...?
f(x)/(x-3) f(x/3)/(x-9)
\[\frac{f(x)}{x-3} = P(x) + \frac{42}{x-3}\]\[x = \frac{x}{3}\] \[\frac{3\cdot f\left(x/3\right)}{x-9} = P(x) + \frac{42\cdot 3}{x-9}\] I love my solution <3
Okay, got it. My calculation mistake :| @Ackhat
@Ishaan94 More explanation is appreciated :) (sorry... I'm stupid :| )
Beautiful solution Ishaan.
Calli: Division Algorithm
I am not a good teacher, Sorry. What part you didn't understand callisto?
Thank you very much foolformath
\[\frac{3\cdot f\left(x/3\right)}{x-9} = P(x) + \frac{42\cdot 3}{x-9}\] ^ don't know where it comes..
Bezu
:)
What a simple solution!!! Great job @Ishaan94 !!!!!
Substitute \(x=\frac x3 \) in \[ \frac{f(x)}{x-3} = P(x) + \frac{42}{x-3} \]
Oh... Got it!!!! Thanks!!! Lovely solution :)
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. \[\large \implies \frac{f\left(\frac x3\right)}{\frac{x-9}3} = P + \frac{42}{\frac {x -9}3}\]
No, it is beautiful :|
What's 'Bezu'? @Akchat
never mind
ok this is a theorem
The most wonderful thing is that my sister understands it :) Once again, thank you everyone :)

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