Modify the rational-arithmetic package to use generic operations, but change make-rat so that it does not attempt to reduce fractions to lowest terms. Test your system by calling make-rational on two polynomials to produce a rational function
Now add rf to itself, using add. You will observe that this addition procedure does not reduce fractions to lowest terms. We can reduce polynomial fractions to lowest terms using the same idea we used with integers: modifying make-rat to divide both the numerator and the denominator by their greatest common divisor. The notion of greatest common divisor'' makes sense for polynomials. In fact, we can compute the GCD of two polynomials using essentially the same Euclid's Algorithm that works for integers.60 The integer version is
where remainder-terms picks out the remainder component of the list returned by the term-list division operation div-terms that was implemented in exercise 2.91.
exercise 2.91. A univariate polynomial can be divided by another one to produce a polynomial quotient and a polynomial remainder. For example,
Division can be performed via long division. That is, divide the highest-order term of the dividend by the highest-order term of the divisor. The result is the first term of the quotient. Next, multiply the result by the divisor, subtract that from the dividend, and produce the rest of the answer by recursively dividing the difference by the divisor. Stop when the order of the divisor exceeds the order of the dividend and declare the dividend to be the remainder. Also, if the dividend ever becomes zero, return zero as both quotient and remainder. We can design a div-poly procedure on the model of add-poly and mul-poly. The procedure checks to see if the two polys have the same variable. If so, div-poly strips off the variable and passes the problem to div-terms, which performs the division operation on term lists. Div-poly finally reattaches the variable to the result supplied by div-terms. It is convenient to design div-terms to compute both the quotient and the remainder of a division. Div-terms can take two term lists as arguments and return a list of the quotient term list and the remainder term list. Complete the following definition of div-terms by filling in the missing expressions. Use this to implement div-poly, which takes two polys as arguments and returns a list of the quotient and remainder polys.
