Algebra: How to factor a trinomial Perfect Square

From your previous topics , You learned that the square of a binomial is a trinomial perfect square. Thus , the factors of a trinomial perfect square are two binomials which may be written as the square of a binomial.

To illustrate this point , let us consider the following examples.
Example.1
Multiplication

factoring

 is the square root of  
while 3 is the square root of 9


Example 2
Multiplication

factoring

  is the square root of 
while 5 is the square root of 25


Note:
From the examples , it is clear that the following points should be noted in factoring a trinomial perfect square.
1.The factors consists pf like binomials whose terms are the square roots of the terms which are perfect squares.
2. The sign connecting the terms of the binomial factor is the same as the sign of the remaining term.


Hence if a trinomial is perfect square, its factors can be easily be determined.
The problem then is , " How do we know that a trinomial is a perfect square?"


let us consider the trinomial 

Analysis: 

or

and

or

 therefore 

and 9 are perfect squares . 2x is called the square root of 

 and 3 is the square root of 9. Now   

and if their product is doubled , we get  

which is the middle term of the trinomial.

Therefore, trinomial is a perfect square if the following conditions exist:
1. Two terms are positive perfect squares.
2. The third term (generally the middle term) is twice the product of the square roots of the terms which are perfect and may be either positive or negative.