# 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.

## Arithmetic Operations

The real numbers have the following properties:

(Commutative Law) | |

(Associative Law) | |

(Distributive law) |

In particular, putting in the Distributive Law, we get

and so

**EXAMPLE 1**

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

or

Similarly, we obtain

**EXAMPLE 2**

## Fractions

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:

**EXAMPLE 3**

## Factoring

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.

Therefore

**EXAMPLE 5 **Factor

SOLUTION Even though the coefficient of
is not 1, we can still look for factors of
the

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 6**

**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
a factor

of

**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:

Therefore

## 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

**EXAMPLE 10**

## Quadratic Formula

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

are

**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

negative:

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

## Radicals

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.)

**EXAMPLE 14**

Notice that because
indicates the positive square root.

(See Appendix A.)

In general, if is a positive integer,

Thus are not defined.
The following

rules are valid:

**EXAMPLE 15**

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

## Exponents

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

the power.

**EXAMPLE 17**

## Exercises

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

the expression.

101–108 Rationalize the expression.

109–116 State whether or not the equation is true for all

values of the variable.