# Unit # 3 - Algebraic
Expression and Manipulation

## Key points

a (b + c) = ab + ac

(a + b)^{2} = a^{2} + 2ab
+ b^{2}

(a − b)^{2} = a^{2} − 2ab
+ b^{2}

(a + b)(a − b) = a^{2} − b^{2}

## Factorization of
Algebraic Expressions

### Taking out a Common
factor

To factories an algebraic expression,
always look for a common factor. If there is a common factor, then take it out
and use the difference of two squares formula.

### Making Groups
(factorize four terms)

2x² - 2x + 5x - 5

= 2x (x - 1) + 5(x - 1)

= (x - 1) (2 x + 5)

This does two things. First, the four
terms are swapped around and regrouped if necessary, then the pairs of terms
are factorized in such a way that a common factor result. In this example, (x-1)
is now a common factor, so that if the factories command is used one more time
the expression will be fully factorized.

### Quadratic Expression

It is a second degree equation, means it
has one variable and the maximum power of variable is 2. It is written as ax^{2}
+ bx + c = 0, where a, b, c are real numbers and a should not be zero. (if a is
zero then it is not a quadratic equation , but a linear equation). Also the
powers of variable should be integer and must not be any other value but 0, 1
and 2.

### Factorizing Quadratic
Expression (Mid-Term Break)

In his method we need to break the middle
term “b” into to numbers, where are the sum of these two numbers should be
equals to b, and their product should be equals to c. then actorise it using
common factor method.

### Quadratic formula

This is very efficient and simple method to find the real and imaginary roots o any quadratic equation. The formula is;

x = (-b + sqrt(b^{2} - 4ac))/2a

and

x = (-b - sqrt(b^{2} - 4ac))/2a

### Evaluation of Algebraic
Expressions

The process of replacing the variables in
an expression with the numerical values and simplifying it is known as
evaluating an algebraic expression. Following order of operation is used to
evaluate an algebraic expression;

1.
Perform
the operations inside a parenthesis first

2.
Then
exponents

3.
Then
multiplication and division, from left to right

4.
Then
addition and subtraction, from left to right