Section A-4: Algebraic Expressions and Operations
The essential concepts of the calculus are few; the manipulative skills needed to successfully put these concepts into use are many. The following examples demonstrate ways to implement some routine operations on algebraic expressions.
The ExpandSteps command in the Student Basics package can provide an annotated stepwise version of some of the algebraic manipulations involved in expanding or simplifying polynomial and rational functions. Its use is illustrated in several of the following examples.
Express 3 x−22 x3+2 x in the form anxn+⋯+a1x+a0, then factor the result.
Obtain the sum of the coefficients of x3 and x4 in the sum of 2 x−a4 x+a2 and 3 x−15.
Express the difference x2−xx3−x−x2−1x2+x as a single fraction.
Obtain the real root of −81/3.
Complete the squares in 3 x2−5 y2+7 x+4 y−9.
From first principles, expand x+y5 by the Binomial theorem.
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