Download Section 9.5 The Binomial Theorem

•
Section 9.5
The Binomial Theorem
163
Name______________________________________________
Section 9.5 The Binomial Theorem
Objective: In this lesson you learned how to use the Binomial Theorem
and Pascal’s Triangle to calculate binomial coefficients and
binomial expansions.
Important Vocabulary
Define each term or concept.
Binomial coefficients The coefficients of a binomial expansion.
I. Binomial Coefficients (Pages 683−684)
What you should learn
How to use the Binomial
Theorem to calculate
binomial coefficients
List four general observations about the expansion of ( x + y ) n
for various values of n.
1) In each expansion, there are n + 1 terms.
2) In each expansion, x and y have symmetrical roles. The powers of x
decrease by 1 in successive terms, whereas the powers of y increase by 1.
3) The sum of the powers in each term is n.
4) The coefficients increase and then decrease in a symmetric pattern.
The Binomial Theorem states that in the expansion of (x + y)n =
xn + nxn – 1y + . . . + nCrxn – ryr + . . . + nxyn – 1 + yn, the coefficient
of xn – ryr is
nCr
= n!/((n − r)!r!)
Example 1: Find the binomial coefficient
792
.
12 C 5 .
II. Pascal’s Triangle (Page 685)
Pascal’s Triangle is a triangular pattern, named for French
mathematician Blaise Pascal, in which the first and last numbers
in each row are 1 and every other number in each row is formed
by . . .
adding the two numbers immediately above the
number.
The numbers in Pascal’s triangle are precisely the same numbers
that are the
coefficients of binomial expansions
.
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to use Pascal’s
Triangle to calculate
binomial coefficients
164
Chapter 9
•
Sequences, Series, and Probability
Construct rows 0 through 6 of Pascal’s Triangle.
1
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
III. Binomial Expansions (Pages 686−687)
Writing out the coefficients for a binomial that is raised to a
power is called
expanding a binomial
.
What you should learn
How to use binomial
coefficients to write
binomial expansions
Example 2: Use the binomial coefficients from the appropriate
row of Pascal’s Triangle to expand ( x + 2) 5
x5 + 10x4 + 40x3 + 80x2 + 80x + 32
Additional notes
Homework Assignment
Page(s)
Exercises
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.