I recently learnt a way to tackle these types of problems - Here I Learnt it, explained to me by @Pranjal Jain

\( f(x) = x^4 + ax^3 + bx^2 + cx + d\)

We can observe that -

\( f(e) = e\times1993\)

thus - \(f(11) - 11\times1993 = 0\) , \(f(-7) + 7\times1993 = 0\)

\( f(11) + f(-7) = 1993(11 - 7) = 1993\times4\)

\(\dfrac{f(11) + f(-7)}{4} = 1993\)

Where I am wrong , please don't provide me the answer , just comment where I am wrong.

Did I started wrongly i.e - \( f(e) = e\times1993\) ?

Thank you

## Comments

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TopNewest\(f(x) \neq 1993x, x \in \mathbb R\) – Krishna Sharma · 2 years ago

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\( f(11) + f(-7) \) – Krishna Sharma · 2 years ago

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– Dewita Sonya · 2 years ago

Why \(c=d=0\)? In my opinion, we can't directly determine the value of \(c\). In fact, we don't have to.Log in to reply

now since c is the root of f(x) thus f(c) = 0 , then d must be zero

Correct me if I am wrong , Thanks for asking the question – Megh Choksi · 2 years ago

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– Dewita Sonya · 2 years ago

I know that you mean c is a root of f(x). But we still don't have to determine its value. I don't really understand why d must be zero if f(c)=0. I agree with Krishna SharmaLog in to reply

– Siddhartha Srivastava · 2 years ago

You only have three equations , \( f(0) = d \) , \( f(c) = 0 \) and \( 6c = d\). How do you get \( d = 0 \)?Log in to reply

Thank you! because of your note I also got a chance to learn . – Shivamani Patil · 2 years ago

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What is your reason behind the claim f(e)=1993e – Agnishom Chattopadhyay · 2 years ago

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– Megh Choksi · 2 years ago

Sorry I misunderstood itLog in to reply

– Agnishom Chattopadhyay · 2 years ago

Wanna know the Wolfram|Alpha approach for this?Log in to reply

– Megh Choksi · 2 years ago

nopeLog in to reply

– Agnishom Chattopadhyay · 2 years ago

Hmm, I wish more people had read A=B and were exposed to EMLog in to reply

– Megh Choksi · 2 years ago

Should I delete this note because in a way I am discussing a problemLog in to reply

– Krishna Sharma · 2 years ago

Just delete that solution commentLog in to reply

Hey! You misunderstood it!! See. \(f(x)-1993x\) has roots \(1, 2\) and \(3\). So we can say that \[f(x)-1993x=(x-1)(x-2)(x-3)(x-\alpha)\] (Since its of degree 4).

\(f(e)=e\) is for \(e=1,2,3\) only.

Carry on from here. I hope you got it this time. – Pranjal Jain · 2 years ago

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– Megh Choksi · 2 years ago

I solved it see a comment is deleted after krishna's comment there i wrote the solution , thanks for helpingLog in to reply

@Calvin Lin As you see in the note, tag didn't worked! I didn't got any mail. Is there any bug? It happens a lot with me – Pranjal Jain · 2 years ago

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– Calvin Lin Staff · 2 years ago

@ mention currently only works in comments. It currently does not work in notes or problemsLog in to reply

– Pranjal Jain · 2 years ago

Sometimes it also causes trouble in comments. Lemme try here. @Pranjal JainLog in to reply