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 and SimplifySteps commands 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. If the package has been loaded, the functionality of the ExpandSteps command is available via the Context Panel. The use of these commands 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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