A70
Appendix A
Review of Fundamental Concepts of Algebra
What
you should learn
•
Avoid common algebraic
errors.
•
Recognize and use algebraic
techniques that are common
in calculus.
Why
you should learn it
An efficient command of algebra
is critical in mastering this course
and in the study of calculus.
Errors and the Algebra of Calculus
A.7
Algebraic Errors to Avoid
This section contains five lists of common algebraic errors: errors involving
parentheses, errors involving fractions, errors involving exponents, errors involv
ing radicals, and errors involving dividing out. Many of these errors are made
because they seem to be the
easiest
things to do. For instance, the operations of
subtraction and division are often believed to be commutative and associative.
The following examples illustrate the fact that subtraction and division are neither
commutative nor associative.
Not commutative
Not associative
20
4
2
20
4
2
15
5
5
15
8
6
2
8
6
2
4
3
3
4
Errors Involving Parentheses
Potential Error
Correct Form
Comment
Change all signs when distributing minus sign.
Remember the middle term when squaring binomials.
occurs twice as a factor.
When factoring, apply exponents to all factors.
3
2
x
2
2
3
x
6
2
3
x
2
2
3
x
6
2
3
x
2
2
1
2
ab
4
1
2
a
1
2
b
1
4
ab
1
2
a
1
2
b
1
2
ab
a
b
2
a
2
2
ab
b
2
a
b
2
a
2
b
2
a
x
b
a
x
b
a
x
b
a
x
b
Errors Involving Fractions
Potential Error
Correct Form
Comment
Leave as
Do not add denominators when adding fractions.
Multiply by the reciprocal when dividing fractions.
Use the property for adding fractions.
Use the property for multiplying fractions.
Be careful when using a slash to denote division.
1
x
2
1
x
2
1
2
x
x
1
x
2
1
x
2
1 3
x
1
3
x
x
3
1 3
x
1
3
x
1
3
x
1
3
1
x
1
3
x
1
3
x
1
a
1
b
b
a
ab
1
a
1
b
1
a
b
x
a
b
x
a
1
b
x
ab
x
a
b
bx
a
a
x
b
.
a
x
b
a
x
a
b
Be careful when using a slash to denote division and be
sure to find a common denominator before you add
fractions.
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Appendix A.7
Errors and the Algebra of Calculus
A71
A good way to avoid errors is to
work slowly
,
write neatly
, and
talk to
yourself
. Each time you write a step, ask yourself why the step is algebraically
legitimate. You can justify the step below because
dividing the numerator and
denominator by the same nonzero number produces an equivalent fraction
.
Using the Property for Adding Fractions
Describe and correct the error.
Solution
When adding fractions, use the property for adding fractions:
Now try Exercise 17.
1
2
x
1
3
x
3
x
2
x
6
x
2
5
x
6
x
2
5
6
x
1
a
1
b
b
a
ab
.
1
2
x
1
3
x
1
5
x
2
x
6
2
x
2
3
x
3
Errors Involving Exponents
Potential Error
Correct Form
Comment
Multiply exponents when raising a power to a power.
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