We encountered perfect square trinomials under multiplying polynomials.
Now, we will put them to work while factoring.
Squaring a binomial creates a perfect square trinomial:
(a + b)2
(a - b)2
(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

What we need to do now, is to "remember" these patterns
so that we can be on the look-out for them when factoring.

 
Trisqpic
Notice the Pattern of the middle term:
The middle term is twice the product of the binomial's first and last terms.
(a + b)² middle term +2ab
(a - b)² middle term -2ab
In (a - b), the last term is -b.

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expin1
Factor: x2 + 12x + 36
 

Solution:
Does this fit the pattern of a perfect square trinomial?
Yes. Both x2 and 36 are perfect squares.
And 12x is twice the product of x and 6.

Since all signs are positive, the pattern is (a + b)2 = a2 + 2ab + b2.
Let a = x and b = 6.

Answer:    (x + 6)2 or (x + 6)(x + 6)



expin2
Factor: 9a2 - 6a + 1
 

Solution:
Does this fit the pattern of a perfect square trinomial?
Yes. Both 9a2 and 1 are perfect squares.
And 6a is twice the product of 3a and 1.

Since the middle term is negative, the pattern is (a - b)2 = a2 - 2ab + b2.
Let a = 3a and b = 1.

Answer:   (3a - 1)2 or (3a - 1)(3a - 1)


expin3
Factor: (m + n)2 + 12(m + n) + 36
 

Solution: This is a sneaky one! Do NOT start by removing the parentheses. Look at the pattern, instead.
.
Does this fit the pattern of a perfect square trinomial?
Yes. Both (m + n)2 and 36 are perfect squares.
And 12(m + n) is twice the product of (m + n) and 6.

Since the middle term is positive, the pattern is (a + b)2 = a2 + 2ab + b2.
Let a = (m + n) and b = 6.

Answer:  ((m + n) + 6)2 or (m + n + 6)2

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expin2
Factor: x2 + 4x + 4 using Algebra Tiles
KEY: algebratileskey        See more about Algebra Tiles.
Place the x2 tile, 4 x-tiles and 4 1-tiles in the grid.


at44
Fill the outside sections of the grid with x-tiles and 1-tiles that complete the pattern.

at5

The algebra tiles show that trinomial is a perfect square trinomial, (x + 2)2.


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