Game Development Reference
In-Depth Information
A third form of binomial that occurs commonly involves two expressions that are
characterized by subtraction. Here is an example:
ð
3
x
5
Þð
4
x
3
Þ
3
xð
4
xÞ
3
xð
3
Þ
5
ð
4
xÞð
5
Þð
3
Þ
12
x
2
9
x
20
x þ
15
12
x
2
29
x þ
15
In such expressions, the multiplication of a negative by a negative value generates
a positive value. The middle term, likewise, is negative, because when you create
the values for the middle term, you combine two negative terms.
Sums, Differences, and Squares
Table 8.2 provides a summary of operations involving a few of the most common
forms of binomial expressions. The discussion that follows examines a few
expressions that illustrate the application of these generalized approaches to
working with binomials.
As an illustration of the product of the sum and difference of the same binomial,
consider this expression:
ðm þ
3
Þðm
3
Þ
¼ m
2
3
m þ
3
m
9 The middle terms cancel out
:
¼ m
2
9
The difference of the squares remains
:
Here is an example of the square of the sum of two terms:
ðm þ
3
Þ
2
¼ mðmÞþ
3
m þ
3
m þ
3
ð
3
Þ
¼ m
2
þ
6
m þ
9
Table 8.2
Sums, Differences, and Squares
Item
Discussion
ða bÞða þ bÞ¼a
2
b
2
The product of the sum and the difference of the same two terms consists of
the square of the second term subtracted from the square of the first term.
2
ða þ bÞ
¼ a
2
þ 2ab þ b
2
The square of the sum of two terms consists of the sum of the square of the first
term, twice the product of the two terms, and the square of the second term.
2
¼ a
2
2ab þ b
2
ða bÞ
The square of the difference of two terms consists of the square of the first
term minus twice the product of the first and second terms, plus the square of
the second term.
