TABLE OF CONTENTS There are many expressions that are polynomials in a computational context that are not included in the previous definitions for polynomials. For example, it is reasonable to consider the expression
as a polynomial in x, even though it does not satisfy the definitions in Section 6.1. Indeed, a CAS views this expression as a polynomial when it solves the quadratic equation u=0 for x. In addition, it is reasonable to view the expressions sin 3 ( x )+2 sin 2 ( x )+3 and ( x+1) 3+2( x+1) 2+3 as polynomials in terms of a complete sub-expression (sin ( x ) or ( x+1)). On the other hand, the expression (3sin ( x )) x 2+(2 ln ( x )) x+4 is not a polynomial in x because the coefficients of the powers of x also depend on x.
The next definition includes the more general polynomial expressions given above.
Definition 6.14. (Mathematical Definition) Let c 1, c 2, , c r be al gebraic expressions and let x 1 , x 2 , , x m be algebraic expressions that are not integers or fractions. A general monomial expression (GME) in { x 1, x 2, , x m} is an expression of the form
where the exponents n j are non-negative integers and each...
