Sometimes, we can get a question like: Find a polynomial, such that.

$f(7)=89183,&space;0&space;leq&space;a_i&space;<&space;7$

Basically, the question asks us for the polynomial that will give as 89183, given that the coefficients are less than 7 and bigger than or equal to 0. Let’s illustrate this:

$f(7)=a*7^5+b*7^4+c*7^3+d*7^2+e*7^1+f$

Please consider that there might be more or less coefficients, this is just to facilitate the interpretation.

Probably, you might already see, that the number 89183 is in radix 10, and the number, using the coefficients, is in radix 7$(abcdef)_7$. When we used our polynomial to obtain the value of it, when x=7, we actually converted a number to radix 10.

Therefore, if you convert the result we got in radix 10 to radix 7, the number we get is the answer to this polynomial problem!

$89183_1_0=521003_7$

Now, we use these coefficients to construct the polynomial:

$f(x)=5x^5+2x^4+x^3+3$

Conclusion:

$f(x)=y$
$y_x=(a_na_n_-_1...a_1a_0)_x$

where $a_i$ are the coefficients of the polynomial.

/Artem