Simplification of expression is one type of algebraic expression solver. It is always preferable to keep an algebraic expression as short as possible. In many cases an algebraic expression can be modified to an easier or simpler form so that it is presentable in a better way. There are many processes by which this can be achieved. The general term used to modify an algebraic expression into a simpler form is called as simplify the algebraic expression. Factoring is converting a given expression into a product of two or more factors. In a rational expression many times you may find common factors between the numerator and denominator and those factors can be cancelled. It becomes an additional simplification of the original expression.


Factor and Simplify the Algebraic Expression

Factoring is converting a given algebraic expression into a product of two or more factors. Thus it is a method to simplify the algebraic expression. The first exercise in factoring is to find the common elements, to the maximum possible extent,  in all the terms of the algebraic expression. In other words, to find the greatest common factor of all the terms. One must have sound algebraic knowledge to find greatest common factors. An algebraic expression can be simplified by so many methods. Algebraic operations are also useful in simplifying algebraic expressions. Such types of simplification are easy and straight forward. But the most important way of simplifying is by factoring.
Many algebraic identities help us in factoring and simplifying algebraic expressions directly.

Some important identities are:
1. x2 – y2= (x + y)(x – y)

2. x2 ± 2xy + y2 = (x ± y)2

3. x3 + y3= (x + y)( x2 – xy + y2)

4. x3 – y3= (x - y)( x2 + xy + y2)

Solved Examples

Question 1: Simplify $\frac{x^2 + 5x - 36}{x^2 - 2x - 8}$
Solution:
Given $\frac{x^2 + 5x - 36}{x^2 - 2x - 8}$

Step 1:

Factorized the numerator and denominator

x2 + 5x - 36 = x2 + 9x - 4x - 36

= x(x + 9) - 4(x + 9)

= (x - 4)(x + 9)

=> x2 + 5x - 36 = (x - 4)(x + 9)

and x2 - 2x - 8 = x2 - 4x + 2x - 8

= x(x - 4) + 2(x - 4)

= (x + 2)(x - 4)

=> x2 - 2x - 8 = (x + 2)(x - 4)

Step 2:


=> $\frac{x^2 + 5x - 36}{x^2 - 2x - 8}$ = $\frac{ (x - 4)(x + 9)}{ (x + 2)(x - 4)}$

= $\frac{ x + 9}{ x + 2}$

=> $\frac{x^2 + 5x - 36}{x^2 - 2x - 8}$ = $\frac{ x + 9}{ x + 2}$. answer
 

Question 2: Solve for x, x2 – 3x + 2

Solution:
Step 1:
Given quadratic expression  x2 – 3x + 2

There is no any common factor.

Step 2:
Split the middle term – 3x  as, – 2x – x. That is,

=> x2 – 3x + 2 = x2 – 2x – x + 2.

Step 3:
Now find ‘x’ is a common factor for the first two terms and -1 as the common factor for the last two terms.
By partial factoring

=> x2 – 2x – x + 2 = x(x – 2) – 1(x – 2).

Step 4:
Now we see the expression (x – 2) is common factor and using that as a factor,

=> x(x – 2) – 1(x – 2) = (x – 2)(x – 1).

Thus, we ultimately find that  x2 – 3x + 2 can be factored and simplified as   
 (x – 2)(x – 1).

=> x2 – 3x + 2 = (x – 2)(x – 1). answer 
 

Question 3: Simplified by factoring, $\frac{x^2 – 3x + 2}{x^2 – 4}$.
Solution:
Given
$\frac{x ^2 – 3x + 2}{x^2 – 4}$ is the given rational expression.

Step 1:

Factorized the numerator and denominator

=> x2 - 3x + 2 = x2 - 2x - x + 2

= x(x - 2) - (x - 2)

= (x - 1)(x - 2)

=> x2 - 3x + 2 = (x - 1)(x - 2)

and x2 - 4 = x2 - 22

= (x - 2)(x + 2)

[ a2 - b2 = (a - b)(a + b) ]

=> x2 - 4 = (x - 2)(x + 2)

Step 2:


$\frac{x^2 - 3x + 2}{x^2 - 4}$ = $\frac{(x - 1)(x - 2)}{(x - 2)(x + 2)} $

= $\frac{ x - 1}{x + 2}$

With an assumption that x ≠ 2, we can cancel the common term (x – 2).

=> $\frac{ x^2 – 3x + 2}{x^2 – 4}$ = $\frac{ x - 1}{x + 2}$. answer