Factoring a Difference of Squares Lesson 30 1 by UdE9fJ

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									           Perfect Squares
• A number times itself gives you a perfect
  square.

• Example: 1 x 1 = 1
                          These are all
•          2x2=4          examples of
                          perfect squares.
•          3x3 =9
•          4 x 4 = 16
       Difference of Squares
• Difference means to subtract.
• Squares – are Perfect squares

• Therefore you should have one Perfect
  Square minus another square

• When you have two binomials with the
  same variables and different signs your
  solutions will be a difference of squares.
Examples Of Difference of Squares
• (y + 2)(y – 2) = y2 – 4

                  Each term is a square



• (a + b)(a – b) = a2 – b2
                        Each term is a square


• (2x – y)(2x + y) = 4x2 – y2
                                     Each term is a square
  Factor a Difference of Squares
• To factor a Difference of Squares, write
  each term as a square:
Example: Factor 25a2 – 16b2


SOLUTION: 25a2 – 16b2 = (5a)2 – (4b)2
                       = (5a – 4b)(5a + 4b)

                                              The other
  The factor is a                             factor is a
  difference of                               sum of the
  the two terms.                              two terms
    Factor a Difference of Squares

     Factor: 36x2 – 49y2

     SOLUTION:
                   36x2 – 49y2
                 = (6x)2 – (7y)2
                 = (6x – 7y)(6x + 7y)



                                        The other
One factor is                           factor is the
the difference                          sum of the
of the two                              two terms
terms
    Factor a Difference of Squares

     Factor: 144x2 – 81y2                               YOU TRY
     SOLUTION:
                  144x2 – 81y2
                 =(   )2 – (   )2
                 =(      )(         )



                                        The other
One factor is                           factor is the
the difference                          sum of the
of the two                              two terms
terms
    Factor a Difference of Squares

     Factor: 144x2 – 81y2                             SOLUTION
     SOLUTION:
                 144x2 – 81y2
             = (12x)2 – (9y)2
             = (12x – 9y)(12x + 9y)



                                      The other
One factor is                         factor is the
the difference                        sum of the
of the two                            two terms
terms
    Factor a Difference of Squares

     Factor: 9 – y2                                      YOU TRY
     SOLUTION:
                  9 – y2
                 =(   )2 – (    )2
                 =(        )(        )



                                         The other
One factor is                            factor is the
the difference                           sum of the
of the two                               two terms
terms
    Factor a Difference of Squares

     Factor: 9 – y2                                 SOLUTION
     SOLUTION:
                   9 – y2
                 = (3)2 – (y)2
                 = (3 - y)(3 + y)



                                    The other
One factor is                       factor is the
the difference                      sum of the
of the two                          two terms
terms
           Factor Completely
To factor the sometimes requires that you do
more than one step process.


Just like in prime factorization where you keep
breaking down the numbers until you have all
prime factors, you do the same in factoring an
expression completely.
                 Factor Completely
 x4 – y4 = (x2)2 – (y2)2
                = (x2 – y2)(x2 + y2)
                = (x)2 – (y)2(x2 + y2)
This is still
a difference
of squares      = (x – y)(x + y)(x2 + y2)
so we need
to factor
again.
                 Factor Completely
 16x4 – 16y4 = (4x2)2 – (4y2)2
                = (4x2 – 4y2)(4x2 + 4y2)
                = (2x)2 – (2y)2(4x2 + 4y2)
This is still
a difference
of squares      = (2x – 2y)(2x + 2y)(4x2 + 4y2)
so we need
to factor
again.
YOU TRY
                    Factor Completely
 81a4 – b4 = (           )2 – (   )2
                =(        )(           )
                =
This is still
a difference
of squares      =
so we need
to factor
again.
SOLUTION
                 Factor Completely
 81a4 – b4 = (9a2)2 – (b2)2
                = (9a2 – b2)(9a2 + b2)
                = (3a)2 – (b)2(9a2 + b2)
This is still
a difference
of squares      = (3a – b)(3a + b)(9a2 + b2)
so we need
to factor
again.
          Factor Completely
(x + 3)2 – y2 = (x + 3 – y)(x +3 + y)

      There is no
      difference
      of squares
      here so we
      are done
         Factor Completely
(x - 2)2 – (y – 6)2
= [(x – 2) + (y – 6)][(x – 2) – (y - 6)]
= (x – 2 + y – 6)(x – 2 - y + 6)
= (x + y – 8)(x – y + 4)
              Class work
• Make sure you have completed Lesson 30
  and check your solutions in the share
  folder or blog.

• Complete Lesson 30(1) worksheet

• Reminder that solutions are in the share
  folder or blog.

								
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