Unit # 3 - Algebraic Expression and Manipulation

Key points

a (b + c) = ab + ac

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

(a + b)(a − b) = a² − b²

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 ax² + 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(b² - 4ac))/2a

and 

x = (-b - sqrt(b² - 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