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 :|*

- Callisto

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- schrodinger

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- anonymous

I thought someone was typing an answer.

- cwtan

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

- Callisto

The answer is not -4 :|

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## More answers

- cwtan

younger sister? it's complicated... LOL

- Callisto

Younger sister...

- anonymous

p+q=6?

- Callisto

Nope :|
p+q=15

- cwtan

Big trouble

- anonymous

oh yes 15

- anonymous

i know wait

- anonymous

:) why my answer is 42

- Callisto

It is!!!!

- anonymous

ohhohohoh

- anonymous

Lol this question is funny.

- Callisto

How did you get that answer? @Ackhat

- 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)

- anonymous

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

- Callisto

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

- anonymous

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

- anonymous

:D

- anonymous

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

- 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 ...?

- anonymous

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

- anonymous

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

- Callisto

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

- Callisto

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

- anonymous

Beautiful solution Ishaan.

- anonymous

Calli: Division Algorithm

- anonymous

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

- anonymous

Thank you very much foolformath

- 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..

- anonymous

Bezu

- anonymous

:)

- cwtan

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

- anonymous

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

- Callisto

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

- anonymous

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.
\[\large \implies \frac{f\left(\frac x3\right)}{\frac{x-9}3} = P + \frac{42}{\frac {x -9}3}\]

- anonymous

No, it is beautiful :|

- anonymous

What's 'Bezu'? @Akchat

- anonymous

@Ackhat *

- anonymous

never mind

- anonymous

ok this is a theorem

- Callisto

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

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