Polynomials

Let $Q(x)$ be the quotient when $x^{13}+x^7$ is divided by $(x-1)^2$ . Then, $x^{13}+x^7=(x-1)^2 Q(x) + P(x) \\\implies (1+x)^{13}+(1+x)^7=x^2Q(x+1)+P(x+1) \\\implies P(x+1)=2+20x \\\implies P(x)= 20x-18 \\\implies P \left(\frac{2033}{20} \right) = \boxed{2015}$

In the transition from the second line to the third, binomial theorem has been used.

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Alternate Solution:

Note that $P(x)$ is a linear polynomial.

Consider the following equation: $x^{13}+x^7=(x-1)^2 Q(x) + P(x)$

Putting $x=1$ , we see that $P(1)=2$ .

Now, differentiating the above equation and then putting $x=1$ , we see that $P'(x)=P'(1)=20$ .

From this much information on $P(x)$ , we conclude that $P(x)=20 x-18$

$x^{13} + x^7\\ =\color{#D61F06}{x^{13}} \color{#20A900}{- 2x^{12}} \color{#CEBB00}{+ x^{11}} \color{#20A900}{+ 2x^{12}} \color{#CEBB00}{- 4x^{11}} \color{#EC7300}{+ 2x^{10}} \color{#CEBB00}{+ 3x^{11}} \color{#EC7300}{- 6x^{10}} \color{#3D99F6}{+ 3x^9} \color{#EC7300}{+4x^{10}} \color{#3D99F6}{- 8x^9} \color{#69047E}{+ 4x^8} \color{#3D99F6}{+5x^9} \color{#69047E}{- 10x^8} \color{#E81990}{+5x^7} \color{#69047E}{+6x^8} \color{#E81990}{- 12x^7} \color{teal}{+ 6x^6} \color{#E81990}{+ 8x^7} \color{teal}{- 16x^6} \color{cyan}{+ 8x^5} \color{teal}{+ 10x^6} \color{cyan}{- 20x^5} \color{#624F41}{+ 10x^4} \color{cyan}{+ 12x^5} \color{#624F41}{-24x^4} \color{magenta}{+12x^3} \color{#624F41}{+14x^4} \color{magenta}{- 28x^3} \color{olive}{+14x^2} \color{magenta}{+ 16x^3} \color{olive}{- 32x^2} \color{#BBBBBB}{+ 16x} \color{olive}{+ 18x^2} \color{#BBBBBB}{- 36x} + 18 \color{#BBBBBB}{+20x} - 18\\ =x^{11}(x^2-2x+1)+2x^{10}(x^2-2x+1)+3x^9(x^2-2x+1)+4x^8(x^2-2x+1)+5x^7(x^2-2x+1)+6x^6(x^2-2x+1)+8x^5(x^2-2x+1)+10x^4(x^2-2x+1)+12x^3(x^2-2x+1)+14x^2(x^2-2x+1)+16x(x^2-2x+1)+18(x^2-2x+1) + 20x-18\\ =(x^{11} + 2x^{10} + 3x^9 + 4x^8 + 5x^7 + 6x^6 + 8x^5 + 10x^4 + 12x^3 + 14x^2 + 16x + 18)(x-1)^2 + 20x-18$

Therefore,

$\dfrac{x^{13} + x^7}{(x-1)^2} = (x^{11} + 2x^{10} + 3x^9 + 4x^8 + 5x^7 + 6x^6 + 8x^5 + 10x^4 + 12x^3 + 14x^2 + 16x + 18)+ \dfrac{20x-18}{(x-1)^2}$

$P(x) = 20x-18\\ P\left(\dfrac{2033}{20}\right) = 20\left(\dfrac{2033}{20}\right) - 18 = 2033 - 18 = \boxed{2015}$