Converting Polynomials into Binary form

Question

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

Answer 1

Just put the value of $x=10$ then

$$10^3+1=1001$$

Answer 2

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 $x^{16} + x^{15} + x^2 +1$.

Answer 3

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 $$100_2 - 010_2 = 010_2 = 2_{10}$$

Answer 4

For this example: $x^7 + x^5 +1$

1. Look at the degree of the polynomial. In the case of the above example it is 7.

2. Write down all numbers from the degree of the polynomial down to 0. These will correspond to the exponents of $x$ in the polynomial. In our case:

   7 6 5 4 3 2 1 0

3. And, finally write a 1 under each exponent for which that power of $x$ occurs in the polynomial, and 0 under the others. 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 power of $x$ with a 1, and replacing the powers that do not occur with a 0.

Note: The last digit stands for 1 because that is $2^0$.