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How can a polynomial such as $x^3 + 1$ be converted to its binary form of $1001$. Likewise, $10100001 = x^7 + x^5 + 1.$

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  • $\begingroup$ What do you mean by "the binary form of a polynomial"? $\endgroup$ – David Richerby Nov 2 '18 at 16:42
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Just put the value of $x=10$ then

$$10^3+1=1001$$

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    $\begingroup$ What if polynomial is like $x^2 -x $ ? $10^2 -10 = 90 $? $\endgroup$ – aaag Mar 11 '17 at 6:20
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The polynomial notation is a shortcut to write binary code while omitting the zeros, it's useful to crunch CRC communication checksum to verify electric signal quality with an XOR comparison operation. Then a binary code like 1 1000 0000 0000 0101 may be noted $x^{16} + x^{15} + x^2 +1$.

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For converting polynomial to its binary form $$p(x) = x^{3} + 1$$ you have to first reduce the coefficients mod 2. This gives us $$x^{3} + 1$$ Now simply substitute $x=2\;$ and evaluate, this gives the d+1 bit number(where d is degree of the polynomial):

$$1000_2 + 0001_2 = 1001_2 = 9_{10}$$

for ex: $$p(x) = x^2-x$$ for each coefficient of p(x) take mod 2 then this gives us : $$p'(x) = x^2-x$$ Now, substitute x = 2 (since degree of polynomial is 2 then it gives us 3(=2+1) bit number) and convert each value to its binary form then evaluate it $$100_2 - 010_2 = 010_2 = 2_{10}$$

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