FACTORING
SPECIAL
CASES
The vocabulary of perfect squares
Perfect squares are numbers like 4, 9, 16, 25, etc.
• Any variable to an even power is a perfect square.
X2, Y4, A6, D8 are all perfect squares
• 4x2 is a perfect square (2x)(2x)
• (3y)2 is a perfect square (3y)(3y)
• The word ‘difference’ means ‘subtract’
The difference of perfect squares
X2 – 9 (difference of perfect squares)
AC number is (1)(9) = 9
B = 0 (no x term) - 9 SUBTRACT
You need 2 numbers that multiply to give 9
and subtract to give 0 ( + 3 and – 3 )
X2 + 3x – 3x – 9 = x(x + 3) – 3(x + 3)
(x + 3)( x – 3 )
Shortcut!
X2 – 9 = (x + 3)( x – 3 )
+ square root 2nd
Square root 1st - square root 2nd
Square root 1st
Pattern is always the same
More examples
4X2 - 9 = (2x + 3)(2x – 3)
25x2 – 64y2 = (5x + 8y)(5x - 8y)
36x2 – 144 = 36(x2 – 4) = 36(x+2)(x-2)
See how much easier it is when you factor
out the greatest common factor first!
The sum of perfect squares
Unless you can pull out a GCF,
they are PRIME
4X2 + 9 Prime
25x2 + 64y2 = Prime
36x2 + 144 = 36(x2 + 4)
See how much easier it is when you factor
out the greatest common factor first!
Perfect square trinomials
Look for the pattern:
4X2 + 12x + 9
4X2 is a perfect square (+2x)(+2x)
9 is a perfect square (+3)(+3)
(+2x)(+3) = +6x doubled = +12x
4X2 + 12x + 9 = (2x+3)(2x+3)
Watch your signs!
4X2 - 12x + 9
4X2 is a perfect square (+2x)(+2x)
9 is a perfect square (-3)(-3)
(+2x)(-3) = -6x doubled = -12x
4X2 - 12x + 9 = (2x-3)(2x-3)
Look for the pattern!
4X2 - 12x + 9
First and last are perfect squares
Middle is double the product of their roots
Last number is always positive
Signs in the parentheses match middle sign
4X2 - 12x + 9 = (2x-3)(2x-3)
4X2 + 12x + 9 = (2x+3)(2x+3)
Fallback plan
These special cases are faster to do if you recognize
them and remember the ‘formula’.
They can also be solved by the same grouping method
that we have used for all other trinomials.
4X2 - 9 = (2x + 3)(2x - 3)
4X2 + 9 PRIME
4X2 - 12x + 9 = (2x - 3)(2x - 3)
4X2 + 12x + 9 = (2x + 3)(2x + 3)
Difference of perfect square
Sum of perfect squares
Perfect square trinomials
You can write the answer just by
looking at them if you can:
recognize them
remember the formula
They also work out just fine if
you do them the old-fashioned
way ~ it just takes longer.