Transcript 7.6 Factoring Differences of Squares

7.6 Factoring
Differences of
Squares
CORD Math
Mrs. Spitz
Fall 2006
Objectives
• Identify and factor polynomials that are
the differences of squares.
Assignment
• pp. 278-279 #4-40 all
• Mid-chapter Review pg. 280
• Quiz B
Application
• Every Friday in Ms. Spitz’s CORD math class,
each student is given a problem that must be
solved without using paper or pencil. This
week Justin’s 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?
Application
• Justin noticed that this product could be
written as the product of a sum and a
difference.
93  87  (90  3)(90  3)
Application
• He then did the calculation mentally
using the rule for this special product.
(90  3)(90  3)  90  3
2
2
 8100  9
 8091
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.
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
a
a
The area a2 – b2 can
b be found by
subtracting the area
of the smaller
square from the
area of the larger
square.
Difference of Squares
a2 – b2 = (a – b)(a + b) = (a + b)(a – b)
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)
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)
1 2 4 2
Ex. 3: Factor t  p
4
9
• You can use this rule to factor trinomials that
can be written in the form a2 – b2.
1 2 4 2
1 2 2 2
t  p  ( t )  ( p)
4
9
2
3
1
2
1
2
 ( t  p)( t  p)
2 3
2
3
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)
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.
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).