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) = 9B = 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 1st + square root 2nd

Square root 1st - square root 2nd

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 squaresMiddle is double the product of their roots

Last number is always positiveSigns 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 squareDifference of perfect squareSum of perfect squaresSum of perfect squares

Perfect square trinomialsPerfect 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.