Callisto
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 :|*
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FoolForMath
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I thought someone was typing an answer.
cwtan
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TT bluring....... btw i found -4.... I duno whether it is right or wrong...
Callisto
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The answer is not -4 :|
cwtan
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younger sister? it's complicated... LOL
Callisto
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Younger sister...
Ackhat
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p+q=6?
Callisto
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Nope :|
p+q=15
cwtan
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Big trouble
Ackhat
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oh yes 15
Ackhat
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i know wait
Ackhat
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:) why my answer is 42
Callisto
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It is!!!!
Ackhat
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ohhohohoh
Ishaan94
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Lol this question is funny.
Callisto
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How did you get that answer? @Ackhat
ash2326
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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)
Ackhat
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f(x)=2x^3-3x^2+p/3x+q
Callisto
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.... I've thought about it.....I swear......
Ishaan94
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f(x)/(x-3)=k+42/(x-3)
3f(x/3)/(x-9)= k + 3*42/(x-9)
Ishaan94
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:D
Ackhat
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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
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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 ...?
Ackhat
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f(x)/(x-3) f(x/3)/(x-9)
Ishaan94
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3
\[\frac{f(x)}{x-3} = P(x) + \frac{42}{x-3}\]\[x = \frac{x}{3}\]
\[\frac{3\cdot f\left(x/3\right)}{x-9} = P(x) + \frac{42\cdot 3}{x-9}\]
I love my solution <3
Callisto
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Okay, got it. My calculation mistake :| @Ackhat
Callisto
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@Ishaan94 More explanation is appreciated :) (sorry... I'm stupid :| )
FoolForMath
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Beautiful solution Ishaan.
FoolForMath
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Calli: Division Algorithm
Ishaan94
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I am not a good teacher, Sorry.
What part you didn't understand callisto?
Ishaan94
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Thank you very much foolformath
Callisto
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\[\frac{3\cdot f\left(x/3\right)}{x-9} = P(x) + \frac{42\cdot 3}{x-9}\] ^ don't know where it comes..
Ackhat
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Bezu
Ackhat
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:)
cwtan
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What a simple solution!!! Great job @Ishaan94 !!!!!
FoolForMath
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Substitute \(x=\frac x3 \) in \[ \frac{f(x)}{x-3} = P(x) + \frac{42}{x-3} \]
Callisto
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Oh... Got it!!!! Thanks!!!
Lovely solution :)
Ishaan94
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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.
\[\large \implies \frac{f\left(\frac x3\right)}{\frac{x-9}3} = P + \frac{42}{\frac {x -9}3}\]
FoolForMath
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No, it is beautiful :|
Ishaan94
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What's 'Bezu'? @Akchat
Ishaan94
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@Ackhat *
Ackhat
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never mind
Ackhat
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ok this is a theorem
Callisto
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The most wonderful thing is that my sister understands it :)
Once again, thank you everyone :)