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Polynomials

Multiplication of Binomials -- Special Cases

Problems

Problems

Square of a Binomial

To square a binomial, multiply the binomial by itself:
(a + b)2 = (a + b)(a + b)


(a + b)2 = (a + b)(a + b)  
  = a 2 + ab + ba + b 2  
  = a 2 + ab + ab + b 2  
  = a 2 +2ab + b 2  

The square of a binomial is always the sum of:

  1. The first term squared,
  2. 2 times the product of the first and second terms, and
  3. the second term squared.

When a binomial is squared, the resulting trinomial is called a perfect square trinomial.

Examples:

(x + 5)2 = x 2 +2(x)(5) + 52 = x 2 + 10x + 25
(100 - 1)2 = 1002 +2(100)(- 1) + (- 1)2 = 10000 - 200 + 1 = 9801
(2x - 3y)2 = (2x)2 +2(2x)(- 3y) + (- 3y)2 = 4x 2 -12xy + 9y 2

Product of the Sum and Difference of Two Terms

When we multiply two polynomials that are the sum and difference of the same 2 terms -- (x + 5) and (x - 5) for example -- we get an interesting result:


(a + b)(a - b) = a(a) + a(- b) + ba + b(- b)  
  = a 2 - ab + ab - b 2  
  = a 2 - b 2  

The product of the sum and difference of the same two terms is always the difference of two squares; it is the first term squared minus the second term squared. Thus, this resulting binomial is called a difference of squares.

Examples:

(7 - 2)(7 + 2) = 72 -22 = 49 - 4 = 45
(x + 9)(x - 9) = x 2 -92 = x 2 - 81
(2x - y)(2x + y) = (2x)2 - y 2 = 4x 2 - y 2
(3x 2 -2)(3x 2 +2) = (3x 2)2 -22 = 9x 4 - 4
(- y + 5x)(- y - 5x) = (- y)2 - (5x)2 = y 2 -15x 2

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