College Algebra

5. Perfect Square Trinomial

A trinomial is a perfect square if the middle term is twice the product of the square roots
of the first and the third term.

Some expression consists of three terms; two terms that are perfect squares, and third
term which is twice the product of the square roots of the two perfect square terms.
The two perfect square trinomials are positive and negative.


x
2
– 2xy + y
2
= ( x – y )
2
or (x – y) (x – y)


Perfect Square Square of 1
st
2
nd

Trinomial the Difference factor factor
(Negative) of the Square
Root

x
2
+ 2xy + y
2
= ( x + y)
2
or (x + y) (x + y)



Perfect Square Square of 1
st
2
nd

Trinomial the Difference factor factor
(Positive) of the Square
Root

Examples:
a – 2ab + b
2
= (a + b)
2

4x
2
+ 36x + 81 = (2x + 9)
2
x
2
– 10x + 25 = (x – 5)
2

CHAPTER 3
ALGEBRAIC FRACTIONS






Rational expression simple fractions are a quotient of two algebraic expressions or
polynomials.
Objectives
1. Define and give examples of basic algebraic fractions
2. Evaluate algebraic fractions

Types of fractions:

1. Proper fraction is one whose degree of polynomial in the numerator is less than the
degree of the polynomial in the denominator.

Examples: 2 ; x + 1
x + 5 x
2
– 2x + 3

2. Improper fraction is one whose degree of polynomial in the numerator is greater than or
equal to the degree of the polynomial in the denominator.

Examples: x
2
– 1 ; x – 1
x + 1 x + 1

3. Equivalent Fraction is a fraction which has the same value but different in the form.

Examples:

16 = 8 ; ac = a
36 18 bc b

3.1 Reduction of Rational Expressions to lowest terms






It is easy to assess the value of a fraction when it is reduced to lowest terms. This is the
main reason why we always reduce a fraction to lowest terms. An algebraic fraction is said to be
in lowest terms if the numerator and the denominator do not have a common factor except 1.



Procedure:

1. Determine the largest common factor of both numerator and denominator.
2. If the numerator or denominator is a polynomial, factor if necessary.
3. Divide the numerator and denominator by all common factors.

Examples:

4xy = 2y ; x2 – 4 = (x + 2) (x – 2) = x + 2
10xz 5z 2x-4 2(x – 2) 2
Objective
Reduce a rational expression to its lowest term.

As shown in the examples, knowledge of the different techniques of factoring
polynomials is needed to be able to reduce an algebraic fraction to lowest terms.

3.2 Addition and Subtraction of Algebraic Fraction






Addition and subtraction of fractions depend on the kinds of denominator of the fraction
have.

Addition and Subtraction of fraction having a common denominator

To add or subtract fractions having a common denominator, add or subtract the
numerators and place the result over the common denominator.

Examples:
3x + 2x = 5x
4y 4y 4y

3.3 Multiplication of fraction






To multiply fractions, multiply their numerators to obtain the numerator of the product,
and multiply the denominators to obtain the denominator of the product. Factors common to
numerator and denominator should, of course, be divided out.



Example:

ac . a
2
= a
3
c
bd b
2
bd
3


3.4 Division of fraction




Objective
Add and subtract algebraic fractions
Objective
Multiply an algebraic fraction.
Objective
Divide an algebraic fraction.


To find the quotient of two fractions, multiply the numerator by the reciprocal of the
denominator. As with multiplication, the numerator and denominator of each fraction should be
put in factored form, and the result reduced to lowest term.

Examples:

1. 12x
2
+ 7xy – 10y
2
) ÷ 9x
2
– 4y
2

x
2

Solution:

12x
2
+ 7xy – 10y
2
) ÷ 9x
2
– 4y
2

x
2

= (12x
2
+ 7xy – 10y
2
) . x
2

9x
2
- 4y
2

= (3x – 2y) (4x + 5y) x
2

(3x – 2y) (3x + 2y)

= (4x + 5y) x
2

(3x + 2y)

2. x
2
(xy – y) . x
2
– y
4
÷ x
6
– x
5
y
2

3x
2
+ 6xy x + y
2
2x + 4y

Solution:

x
2
(x – y) . x
2
– y
4
÷ x
6
– x
5
y
2

3x
2
+ 6xy x + y
2
2x + 4y

= xy(x – 1) . (x –y
2
) (x + y
2
) . 2 (x + 2y)
3x(x + 2y) x + y
2
x
5
(x – y
2
)

= 2y (x
2
– 1)
3x
4

CHAPTER 4
EXPONENTS AND RADICALS

4.1 Exponent





Objective
Apply the laws of exponents in simplifying algebraic expressions.

Definition of Terms

 Exponent is the number written on the top right side of a quantity to indicate how many times
that the quantity is used as a factor in multiplication. Thus, a
4
, 4 is the exponent.

 Factor is any one of two or more quantities that are multiplied. Thus, in axb = ab, a and b are
factors.

 Power of a quantity is the product obtained when a quantity is multiplied by itself one or
more times.

 Base of a power is the quantity that is multiplied by it one or more times. Thus, in 5
2
5 is the
base, 2 is the exponent; the product of 5 x 5 = 25 is the power. 5
2
are read “5 squared” or “5
raised to the second power”.

 Square root of a quantity is one of its two equal factors. The quantity of 36 has two square
roots +6 and -6. The position square quantity or number has two square roots.

Laws of Exponents

1. The Multiplication Law

To multiply two expressions having the same base, retain the base and add the
exponents. That is a
m
. a
n
= a
m+n

Examples:

3 .3
4
= 3
1+4
= 3
5

4
2
.4 = 4
2+1
= 4
3

2. The Division Law

The quotient of a power with the same base is equal to the base raised to an
exponent equal to the difference of the exponents of the numerator and denominator, that
is:


Examples:

a.
m
/a
n
= a
m-n
if m > n
4
5
/4
3
= 4
5-3
= 4
2
= 16

b. a
n
/a
n
= 1 if m < n
a
n
-a
n


c. a
n
/a
n
= a
0
= 1 if m = n

Any quantity (Except 0) with the zero exponents has a numerical value of 1.

Example 5
0
= 1

3. The Power of a Product

If a product is raised to a power, the result equals the product of each factor raised
to that power that is (a . b) a
n
. b
n

Example:
(3. 2)
4
= 3
4
. 2
4
= 81 . 16 = 1296

4. The Power of a Quotient

The power of a quotient is equal to the quotient of their powers, that is (a/b)
n
=
a
n
/b
n

Example:
(2/3)
4
= 2
4
/3
4
= 16
81
5. The Power of a Power

If an expression containing a power is raised to a power retain the based and
multiply the powers, that is (a
m
)
n
= a
mn


Example:
[(-3)
3
]
2
= (-3)
3.2
= -3
6
= 729

6. Rational Exponent

The fractional exponent means that the base is to be raised to a power indicated
by the number of the fraction and the root is to be extracted whose index is the
denominator of fractions. That is am/n =
n
a
m


Examples:
8
2/3
= √

. √

= 4
2.5
3/4
= √

. 5
3
= 125
7. Negative Exponent

Any quantity with a negative exponent is equal to the reciprocal of the quantity
with the corresponding positive exponent. That is, a
-m
= 1/a
m


Example:
3
-3
= 1 = 1
3
2
9