Let P(x) be a polynomial with real coefficients such that ∫01 xmP(1−x)dx=0∀m∈N∪{0}, then

# Let P(x) be a polynomial with real coefficients such that ${\int }_{0}^{1} {x}^{m}P\left(1-x\right)dx=0\mathrm{\forall }m\in \mathbf{N}\cup \left\{0\right\}$, then

1. A

$P\left(x\right)={x}^{n}\left(1-x{\right)}^{n}$ for some $n\in \mathbf{N}$

2. B

$P\left(x\right)=\left(1-x{\right)}^{2n}$ for some $n\in \mathbf{N}$

3. C

$P\left(x\right)=1-{x}^{m}\left(1-x{\right)}^{n}$ for some $m,n\in \mathbf{N}$

4. D

P(x) = 0

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### Solution:

$0={\int }_{0}^{1} {x}^{m}P\left(1-x\right)dx={\int }_{0}^{1} \left(1-x{\right)}^{m}P\left(x\right)$

Let $P\left(x\right)=\sum _{k=0}^{n} {a}_{k}{x}^{k}$ where ${a}_{k}\in \mathbf{R}$

where

Now ${\int }_{0}^{1} \left(P\left(x\right){\right)}^{2}dx={\int }_{0}^{1} P\left(x\right)\left(\sum _{k=0}^{n} {b}_{k}\left(1-x{\right)}^{k}\right)dx$

$=\sum _{k=0}^{n} {b}_{k}{\int }_{0}^{1} \left(1-x{\right)}^{k}P\left(x\right)dx=0$

As P(x) is a polynomial,  