Prealgebra & Introductory Algebra

Prealgebra & Introductory Algebra

4.5 Factoring Perfect Square Trinomials and the Difference of Two Squares Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Perfect Square Trinomials A trinomial that is the square of a binomial is called a perfect square trinomial. Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 2 Perfect Square Trinomials In the last chapter we learned a shortcut for squaring a binomial (a + b)2 = a2 + 2ab + b2 (a b)2 = a2 2ab + b2 We can now use these special products to help us factor perfect square trinomials, by reversing the equations. Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 3 Factoring Perfect Square Trinomials a2 + 2ab + b2 = (a + b)2 a2 2ab + b2 = (a b)2 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed

4 Factoring Perfect Square Trinomials Example Decide whether 16x2 8xy + y2 is a perfect square trinomial. 1.Two terms, 16x2 and y2, are squares. 16x2 = (4x)2 and y2 = (y)2 2.Twice the product of 4x and y is the opposite of the other term of the trinomial. 2(4x)(y) = 8xy, the opposite of -8xy Thus, 16x2 8xy + y2 is a perfect trinomial square.

16x2 8xy + y = (4x y)2 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 5 Factoring Perfect Square Trinomials Example Factor: 16x2 8xy + y Since we have determined that this trimonial is a perfect square, we can factor it using a special product formula. 16x2 8xy + y = (4x)2 2 4x y + (y)2

a2 2 a b + b2 = (4x y)2 (a b)2 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 6 Difference of Two Squares A binomial is the difference of two squares if 1. both terms are squares and 2. the signs of the terms are different.

For example, 9x2 25y2 c4 + d4 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 7 Difference of Two Squares Factoring the Difference of Two Squares a2 b2 = (a + b)(a b)

Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 8 Difference of Two Squares Example Factor: x2 9. The first term is a square and the last term, 9, is a square and can be written as 32. The signs of each term are different, so we have the difference of two squares.

x2 9 = (x 3)(x + 3). Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 9

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