FACTORING SPECIAL CASES

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FACTORING SPECIAL CASES
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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

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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 )

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Shortcut!

• X2 9 (x 3)( x 3 )
• Square root 1st square root 2nd
• Square root 1st - square root 2nd
• Pattern is always the same

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More examples

• 4X2 - 9 (2x 3)(2x 3)
• 25x2 64y2 (5x 8y)(5x - 8y)
• 36x2 144 36(x2 4) 36(x2)(x-2)
• See how much easier it is when you factor out the
  greatest common factor first!

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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!

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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 (2x3)(2x3)

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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)

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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 (2x3)(2x3)

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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)

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Difference of perfect squareSum of perfect
squaresPerfect 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.