Problem 1 Write a function, called H, which takes as input arguments two polynomials f and g as
symbolic expressions, of equal degree in unknown x, and returns the simplified polynomial h(x) using
steps listed above. Make sure that your code can generate 2N-1 random points in range (-10, 10),
inclusively.
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Note that Sage has an implementation of the Lagrange interpolation but you CANNOT use it – you
must implement your own, based on the formula given above. You are free, however, to use all other
Sage functions we have covered so far in our course.
Illustrate you function H on polynomials f(x)
=27 + 6205
– 2 + 9 and g(x) = 307 – 926 +
– I – 4 by providing an output in order to receive credit. That is, call the function H on the two
polynomials below your code, print the result to screen, to show that your function works. Not producing
an output illustration will result in a loss of points. 30 Points
Problem 2 Write a function, called F, which takes as input arguments two polynomials f and g as
symbolic expressions, of equal degree in unknown . The function F, inside its definition makes a call
to the function H, using the polynomials f and g as input arguments. The function F then returns a list
containing two sublists, where the first sublist contains only the real roots and the second sublist contains
the only imaginary roots (those containing the imaginary unit V-1
= I). Illustrate you function F on
polynomials f(x) = 27 + 6205
– 2+9 and g(x)
-9206 +4205 +203
– 4 by providing list
output that shows the roots in order to receive credit. That is, call the function F on the two polynomials
below your code, print the result to screen, to show that your function works. Not producing an output
illustration will result in a loss of points. 20 Points
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