Review of Algebra
Here we review the basic rules and procedures of algebra
that you need to know in
order to be successful in calculus.
The real numbers have the following properties:
In particular, putting in the Distributive Law, we get
If we use the Distributive Law three times, we get
This says that we multiply two factors by multiplying each
term in one factor by each
term in the other factor and adding the products. Schematically, we have
In the case where and , we have
Similarly, we obtain
To add two fractions with the same denominator, we use the Distributive Law:
Thus, it is true that
But remember to avoid the following common error:
(For instance, take to
see the error.)
To add two fractions with different denominators, we use a common denominator:
We multiply such fractions as follows:
In particular, it is true that
To divide two fractions, we invert and multiply:
We have used the Distributive Law to expand certain
algebraic expressions. We sometimes
need to reverse this process (again using the Distributive Law) by factoring an
expression as a product of simpler ones. The easiest situation occurs when the expression
has a common factor as follows:
To factor a quadratic of the form we note that
so we need to choose numbers so that
EXAMPLE 4 Factor
SOLUTION The two integers that add to give 5 and multiply
to give -24 are -3 and 8.
EXAMPLE 5 Factor
SOLUTION Even though the coefficient of
is not 1, we can still look for factors of
form Experimentation reveals that
Some special quadratics can be factored by using Equations
1 or 2 (from right to
left) or by using the formula for a difference of squares:
The analogous formula for a difference of cubes is
which you can verify by expanding the right side. For a sum of cubes we have
EXAMPLE 7 Simplify
SOLUTION Factoring numerator and denominator, we have
To factor polynomials of degree 3 or more, we sometimes use the following fact.
6 The Factor Theorem If P is a polynomial and
, then is
EXAMPLE 8 Factor
SOLUTION Let where b
is an integer, then
b is a factor of 24. Thus, the possibilities for b are
and . We find that By the Factor Theorem,
is a factor. Instead of substituting further, we use long division as follows:
Completing the Square
Completing the square is a useful technique for graphing
parabolas or integrating
rational functions. Completing the square means rewriting a quadratic
in the form and can be accomplished by:
1. Factoring the number from the terms involving .
2. Adding and subtracting the square of half the coefficient of .
In general, we have
EXAMPLE 9 Rewrite
by completing the square.
SOLUTION The square of half the coefficient of x is 1/4. Thus
By completing the square as above we can obtain the
following formula for the roots
of a quadratic equation
7 The Quadratic Formula
The roots of the quadratic equation
EXAMPLE 11 Solve the equation
SOLUTION With the quadratic formula gives the solutions
The quantity that
appears in the quadratic formula is called the
discriminant. There are three possibilities:
1. If the equation has
two real roots.
2. If the roots are equal.
3. If the equation has no real root. (The roots are complex.)
These three cases correspond to the fact that the number
of times the parabola
crosses the -axis is 2, 1, or 0 (see Figure 1). In case (3) the quadratic
can’t be factored and is called irreducible.
EXAMPLE 12 The quadratic
is irreducible because its discriminant is
Therefore, it is impossible to factor .
The Binomial Theorem
Recall the binomial expression from Equation 1:
If we multiply both sides by and simplify, we get the binomial expansion
Repeating this procedure, we get
In general, we have the following formula.
9 The Binomial Theorem If is a positive integer, then
EXAMPLE 13 Expand
SOLUTION Using the Binomial Theorem with we have
The most commonly occurring radicals are square roots. The
symbol means “the
positive square root of.” Thus
Since , the symbol
makes sense only when
Here are two rules
for working with square roots:
However, there is no similar rule for the square root of a
sum. In fact, you should
remember to avoid the following common error:
(For instance, take to see the error.)
Notice that because
indicates the positive square root.
(See Appendix A.)
In general, if is a positive integer,
Thus are not defined.
rules are valid:
To rationalize a numerator or denominator that contains an
expression such as
we multiply both the numerator and the denominator by the conjugate radical
Then we can take advantage of the formula for a difference of squares:
EXAMPLE 16 Rationalize the numerator in the expression
SOLUTION We multiply the numerator and the denominator by the conjugate radical
Let be any positive number and let be a positive integer. Then, by definition,
11 Laws of Exponents Let and be positive numbers
and let and be any
rational numbers (that is, ratios of integers). Then
In words, these five laws can be stated as follows:
1. To multiply two powers of the same number, we add the exponents.
2. To divide two powers of the same number, we subtract the exponents.
3. To raise a power to a new power, we multiply the exponents.
4. To raise a product to a power, we raise each factor to the power.
5. To raise a quotient to a power, we raise both numerator and denominator to
1–16 Expand and simplify
17–28 Perform the indicated operations and simplify.
29–48 Factor the expression.
49–54 Simplify the expression.
55–60 Complete the square.
61–68 Solve the equation
69–72 Which of the quadratics are irreducible?
73–76 Use the Binomial Theorem to expand the expression.
77–82 Simplify the radicals.
83–100 Use the Laws of Exponents to rewrite and simplify
101–108 Rationalize the expression.
109–116 State whether or not the equation is true for all
values of the variable.