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ParthKohli
 one year ago
Where could I possibly have gone wrong?!
ParthKohli
 one year ago
Where could I possibly have gone wrong?!

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ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Consider all monic quadratic polynomials \(f(x)\) satisfying the following relation:\[f(f(1)) = f(f(0)) = 0\]Find the sum of \(f(3)\) across all such polynomials. Here's my foolproof solution:

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2OK, so let \(f(x) = x^2 + bx + c \) for a second. Now \(f(0) = c \Rightarrow f(f(0)) = f(c) = 0\). This means that \(c\) is a root of this quadratic! Now the other root must be \(1\) so that the product of roots becomes \(c\). Therefore, so far, \(f(x) = (x1)(xc)\). We use the second relation to find \(c\). As you can clearly see, \(f(1) = 2c + 2\) meaning that \(f(2c + 2) = 0\). Thus, \(2c+2\) is also a root of the polynomial  but \(2c+2\) must either be \(c\) or \(1\). We consider both possibilities: 1. \(2c + 2 = 1\Rightarrow c = 1/2\) 2. \(2c + 2 = c\Rightarrow c = 2\) Thus, the two possible quadratics are \((x1)(x+1/2)\) and \((x1)(x+2)\).

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2The answer comes out to be \(2\cdot 3.5 + 2\cdot 5 = \boxed{17}\). Which isn't the answer.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0it looks very good to me! not able to find any mistakes

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2\[f(x) = x^2 + x\]also seems to satisfy this.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2I didn't consider the case \(c=0\) :)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2If \(c=0\), then the polynomial is in the form \(f(x) = x^2 + bx\) so \(f(0) = 0\) and so \(f(f(1))=0 \Rightarrow f(1b) = 0 \Rightarrow (1b)^2 + b(1  b) = 0 \).\[\Rightarrow b^2  2b + 1 + b  b^2 = 0\]\[\Rightarrow b = 1\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2\[f(3) = 9 + 3 = 12 \]so\[s = 29\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Thanks guys  I'm done.
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