7.6 Factoring Differences of Squares

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7.6 Factoring Differences of Squares

• CORD Math
• Mrs. Spitz
• Fall 2006

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Objectives

• Identify and factor polynomials that are the
  differences of squares.

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Assignment

• pp. 278-279 4-40 all
• Mid-chapter Review pg. 280
• Quiz B

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Application

• Every Friday in Ms. Spitzs CORD math class, each
  student is given a problem that must be solved
  without using paper or pencil. This week
  Justins problem was to find the product of 93
  87. How did he determine the answer of 8,091
  without doing the multiplication on paper or
  using a calculator?

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Application

• Justin noticed that this product could be written
  as the product of a sum and a difference.

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Application

• He then did the calculation mentally using the
  rule for this special product.

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Product of sum and difference

• The product of the sum and difference of two
  expressions is called the difference of
  squares. The process for finding this product
  can be reversed in order to factor the difference
  of squares. Factoring the difference of squares
  can be modeled geometrically.

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Product of sum and difference

• Consider the two squares shown below. The area
  of the larger square is a2 and the area of the
  smaller square is b2.

b
The area a2 b2 can be found by subtracting the
area of the smaller square from the area of the
larger square.
b
a
a
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Difference of Squares

• a2 b2 (a b)(a b) (a b)(a b)

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Ex. 1 Factor a2 - 64

• You can use this rule to factor trinomials that
  can be written in the form a2 b2.
• a2 64 (a)2 (8)2
• (a 8)(a 8)

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Ex. 2 Factor 9x2 100y2

• You can use this rule to factor trinomials that
  can be written in the form a2 b2.
• 9x2 100y2 (3x)2 (10y)2
• (3x 10y)(3x 10y)

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Ex. 3 Factor

• You can use this rule to factor trinomials that
  can be written in the form a2 b2.

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Ex. 4 Factor 12x3 27xy2

• Sometimes the terms of a binomial have common
  factors. If so, the GCF should always be
  factored out first. Occasionally, the difference
  of squares needs to be applied more than once or
  along with grouping in order to completely factor
  a polynomial.
• 12x3 27xy2 3x(4x2 9y2)
• 3x(2x 3y)(2x 3y)

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Ex. 5 Factor 162m4 32n8

• 162m4 32n8 2(81m4 16n8)
• 2(9m2 4n4)(9m2 4n4)
• 2(3m 2n2)(3m 2n2)(9m2 4n4)
• 9m2 4n4 cannot be factored because it is not a
  difference of squares.

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The measure of a rectangular solid is 5x3 20x
2x2 8. Find the measures of the dimensions of
a solid if each one can be written as a binomial
with integral coefficients.

• 5x3 20x 2x2 8 (5x3 20x) (2x2 -8)
• 5x(x2 4) 2(x2 - 4)
• (5x 2)(x2 - 4)
• (5x 2)(x 2)(x 2)
• The measures of the dimensions are (5x 2), (x
  2), and (x 2).