# Write a program for addition, multiplication and division with the remainder of

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Write a program for addition, multiplication and division with the remainder of sparse polynomials, i.e. polynomials for which we store only non-zero terms.
A sparse polynomial is specified as a sequence of pairs (e,x), where e is the exponent and x the corresponding coefficient. Pairs are listed in order of decreasing exponents, each on a separate line. The entire polynomial is terminated by the line -1 -1. We never list pairs with zero coefficients, especially the zero polynomial is represented by the line -1 -1 itself.
For example, we write the polynomial 3xˇ5−xˇ2+7 as follows:
5 3
2 -1
0 7
-1 -1
The first line of input tells us what operation to perform: add for addition, mul for multiplication, or div for division with a remainder. Two sparse polynomials are entered on the next lines.
Write the result of the operation again in the form of a sparse polynomial. The result of the div operation is two polynomials: the quotient and the remainder after division. Note that the division may result in odd numbers; list them as fractions (we recommend using the Fraction type for calculations).
For example: I need to work this for general input !!
for input
6 1
4 3
3 -3
0 5
-1 -1
3 3
0 -3
-1 -1
is the correct output
6 1
4 3
0 2
-1 -1
If the operation were mul, the correct output would be:
9 3
7 9
6 – 12
4 -9
3 24
0 -15
-1 -1
And finally, for a surprise, this one:
3 1/3
1 1
0 -2/3
-1 -1
13
0 3
-1 -1
it needs to write all 3 outputs if that is possible
Note: NumPy functions for working with polynomials are not suitable for this task, as they do not use the sparse representation of polynomials. They are therefore too slow for high degree polynomials with many zero terms.

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