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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 $? $\endgroup$ – user35837 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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  • $\begingroup$ What is the polynomial of this? 0xC96C5795D7870F42 Could you help me? $\endgroup$ – Arash Nov 9 '20 at 4:42
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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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For this example: $x^7 + x^5 +1$

  1. You have to see the grade of the polynomial. In the case of the above example it is 7.

  2. Write the grade of the polynomial to 0. This will be the grades of the polynomial. In our case:

    7 6 5 4 3 2 1 0

  3. And, lastly write a 1 under each grade ONLY if it has an $x$. Like this:

    7 6 5 4 3 2 1 0

    1 0 1 0 0 0 0 1

So the result is 10100001. Basically you are replacing each $x$ with a 1, and if it doesn't have an $x$, you replace it with a 0.

Note: Grade 0 is a 1 because it is $2^0$.

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