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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.