$Is it possible to transform $f(x)=x^2+4x+3$ into $g(x)=x^2+10x+9$ by the given sequence of transformations?$

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Is it possible to transform $f(x)=x^2+4x+3$ into $g(x)=x^2+10x+9$ by the transformations

\[f(x) \rightarrow x^2 f(1+\frac{1}{x}) or f(x) \rightarrow (x-1)^2f(\frac{1}{x-1})?\]

Calculus Putnam Competition Math Competitions

Shashank

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Consider the polynomial $P(x)=ax^2+bx+c$ which has the discriminant $b^2-4ac$. The first transformation changes $P$ to $P_1(x)=(a+b+c)x^2+(b+2a)x+a$ which has the discriminant $b^2-4ac$.

The second transformation transforms $P$ to $P_2(x)=cx^2+(b-2c)x+(a-b+c)$ which has the discriminant $b^2-4ac$.

So the discriminant is invarianet under the given two transformations. Since discriminant of $f(x)$ is $4$ and the discriminant of $g$ is 64, $g$ can NOT be obtained from $f$ with those two transformations.

Nirenberg

$Is it possible to transform $f(x)=x^2+4x+3$ into $g(x)=x^2+10x+9$ by the given sequence of transformations?$

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