Binomial Expansions

Math
Reflection
Introduction
Math is the language of science and engineering, it is the tool
to solve problems. Math language and engineering can :
A
binomial is a mathematical expression of two unlike terms
with coefficients and which is raised to at least the power of
1.
Examples of binomials are (
a+b
), (x+2)^2, (a+c)^3, (y

4)^6.
Math
Language
Express
Ideas
Communicating
results
Engineering
Formulate
Analyze
Solving
practical
problems
If you were an engineer 100 years ago,
explain how our method may have been
useful rather than just using long
multiplication?
If I was an engineer 100 years ago, the
formula (a+b)^2 =a^2 +2ab+b^2 and (a

b)^2 =a^2

2ab+b^2 , would save time, long
ago, and also help solve problems such as:
To calculate the labor rates and
percentage of sick days.
Traditional results
analysis methods that can provide an
intuitive approach to valuing projects with
flexible management, or
real
options. 100
years ago, engineers could have used a
binomial conclusion tree with risk

neutral
probabilities
to evaluate what will happen
t
the project overtime.
Binomials
can also be used to model
different situations, like in the stock
market
of engineering companies to
see how
the
prices
will vary over time
. If the engineers
had this tool100 years ago they would have
achieved much more successful projects.
Example, (2)
Since
polynomials (special kind
of binomials) are used to
describe curves of various
types, people use them in the
real world to graph curves. For
example, roller coaster
designers may use polynomials
to describe the curves in their
rides. Combinations of
polynomial functions are
sometimes used in economics
to do cost analyses
. 100 years
ago the engineers could make
the families happier with these,
roller coasters and fun fairs.
Example (3)
Another example if I
was an engineer 100 years
ago, I will use the formula
to design, water canals, the
binomial expansion will
help me calculate the slope,
cross sectional area and
flow rates of the water.
Also the formula will
help me engineer roads, by
calculating the volume of
gravel and asphalt and
multiply the quantity
needed for the whole road.
At What point would our method be big and
cumbersome? (
ie
. Hw many decimal places or
what sorts of numbers would make you think twice
about using this method?)
A binomial is an algebraic expression containing 2 terms.
We sometimes need to expand binomials as follows:
•
(
a
+
b
)
0
= 1
•
(
a
+
b
)
1
=
a
+
b
•
(
a
+
b
)
2
=
a
2
+ 2
ab
+
b
2
•
(
a
+
b
)
3
=
a
3
+ 3
a
2
b
+ 3
ab
2
+
b
3
•
(
a
+
b
)
4
=
a
4
+ 4
a
3
b
+ 6
a
2
b
2
+ 4
ab
3
+
b
4
•
(
a
+
b
)
5
=
a
5
+ 5
a
4
b
+ 10
a
3
b
2
+ 10
a
2
b
3
+ 5
ab
4
+
b
5
This Can be rather difficult for larger powers or more
complicated expressions. And numbers with lots of
decimals.
Alternative methods
•
The coefficients (the numbers in front of each term) follow
a pattern.
•
1 1
•
1 2 1
•
1 3 3 1
•
1 4 6 4 1
•
1 5 10 10 5 1
•
1 6 15 20 15 6 1
•
You can use this pattern to form the coefficients, rather than
multiply everything out as we did above.
•
Our method is good to use if a company is using certain
numbers repeatedly, in her/his data system, but if they have
numbers exceeding 170, it will not be a practical formula.
Can you give us some detailed
examples and explanations of where
long multiplication is more efficient than
our expansion method?
The binomial theorem describes the algebraic expansion of powers of a
binomial. According to the theorem, it is possible to expand the power
(
x
+
y
)
n
into a sum involving terms of the form
ax
b
y
c
, where the coefficient
of each term is a positive integer, and the sum of the exponents of
x
and
y
in each term is
n
. For example, the coefficients appearing in the binomial
expansion are known as binomial coefficients. They are the same as the
entries of Pascal's triangle( the one in the 7
th
slide). The most basic example
of the binomial theorem is the formula for square of
x
+
y
:
(x+y)^2=x^2+2xy+y^2.
But for long multiplication, it is a special method for multiplying larger
numbers. It is a way to multiply numbers larger than 10 that only needs
your knowledge of the ten times Multiplication Table. In that case it is
easier to use the long multiplication. Let us say we want to multiply,
612
×
24:
•
First we multiply
612
×
4
(=2,448),
•
then we multiply
612
×
20
(=12,240),
•
and last we add them together (2,448+12,240=14,688).
Development of Long
Multiplication
Another example of long
multiplication, is this diagonal grid
that was published in Chicago
Tribune, a lot of long multiplication
grids, are being invented and renewed
to make the calculations shorter, and
time consuming.
An example of using long
multiplication is
Genetic Engineering:
The graph shows one use of long
multiplication for DNA sequencing
applications, lab results.
Completely sequenced genomes in Genomes On

Line
Database (GOLD) and Integrated Microbial Genomes
(IMG)
Development of Long
Multiplication
Another application for long
multiplication is
Weather forecasting,
which is an application for science
technology and math, it tells the current
and future of weather and sky conditions. It
also impacts the agriculture, and trade
markets. The nature of the atmosphere
makes it to requires massive power to solve
the equations to solving the atmosphere,
these equations have to be very accurate
since they impact life and property. The
simple long multiplications is part of these
equations.
Conclusion:
These examples prove that the long
multiplication can sometimes be more
efficient than our binomial method. In
Genetic Engineering, it can change the face
of the future, in the Weather Forecast, the
right calculations can save lives and
property.
The data and and calculations helped the
scientists predict the Rita hurricane in 2005.
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