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 middle term +2ab
(a - b middle term -2ab
In (a - b), the last term is -b.

divider

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

divider


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.


divider

NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use".

Topical Outline | Algebra 1 Outline | MathBitsNotebook.com | MathBits' Teacher Resources
Terms of Use
   Contact Person: Donna Roberts

Copyright © 2012-2025 MathBitsNotebook.com. All Rights Reserved.