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8. Using Sage to Solve Problems Symbolically When we say that a computer has solved a problem for us, that can come out to be in one of two flavors: symbolically, or numerically. When we solve numerically, we get a decimal expansion for a (very good) approximation of the answer. When we solve symbolically, we get an exact answer, often in terms of radicals or other complicated functions. 23606797749979 The former is an example of a symbolic solution, and the latter a numerical solution. Often, a numerical answer is desired.

You can feel free to use which ever format you might happen to feel more comfortable with. rref() on its own line, and we learn that the RREF is 1 0 0 0 −107/7 0 1 0 0 −12/7 0 0 1 0 54/7 19/7 0 0 0 1 which gives the final answer of w = −107/7 x = −12/7 y = 54/7 z = 19/7 for the original system of equations. Now we must check our work. For example, we check the first equation with print 2*(-12/7) - 5*(19/7) + 54/7 and print 6 + -107/7 both of which come out to -65/7. The point is not that they come out to -65/7, but rather that they come out equal.

Of course, we knew that from the factorizations that we found earlier. The common factor of (x − 2)(x − 3) and (x − 3)(x − 5) is clearly (x − 3). It should be noted that these examples were trivial because we used quadratic polynomials. The following polynomial is one that I would not want to factor by hand—though, admittedly, the integer roots theorem7 7If f (x) is a polynomial with all integer coefficients, then any integer root must be a divisor of the constant term of f (x). Most numbers have a modest number of divisors, so it is relatively easy to simply check each of them, and thereby have a complete list of all the integer roots of f (x).