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

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