Transcript Pascal`sTriangle

Digital Lesson
(Play the presentation and turn on your volume)
Ms. Nong
 x  a
n
Binomials
 x  a
n
x represent a number or letter
a represent a number or letter
n represents a power
For example: (1 + x)2 =
Questions…
Find
Expand the binomial
or
(x + 2)3 =
Multiply
What is..?
Binomial Theorem
part 1:
Fill in the missing numbers for this triangle on your paper
The Sums of the Rows
The sum of the numbers in any row is equal to 2 to the n th power or 2n, when n is the number of the row. For example:
Describe the patterns
part 1:
20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16
The sum of the numbers in any row is equal to
2 to the nth power or ____
2n when n is
___
the number of the row
___________________.
A binomial is a polynomial with two terms such as x + a. Often
we need to raise a binomial to a power. In this section we'll
explore a way to do just that without lengthy multiplication.
0
x  a 1
Can you see a
1
pattern?


 x  a  x  a
2
2
2
 x  a   x  2ax  a
Can you make a
guess what the next
one would be?
3
2
2
3
x

a

x

3
ax

3
a
x

a


3
 x  a   x  4ax  6a x  4a x  a
5
5
4
2 3
3 2
4
5
x

__
ax

__
a
x

__
a
x

__
a
x

a
x

a



4
4
3
2
2
3
4
We can easily see the pattern on the x's and the a's. But what about the
coefficients? Make a guess and then as we go we'll see how you did.
Let's list all of the coefficients on the x's and the a's and look for a pattern.
 x  a
5
1
 1x  5ax  10a x  10a x  5a 4 x  1a 5
5
4
2 3
3 2
1+ 1
 x  a  1
1
 x  a   1x  1a
2
2
2
 x  a   1x  2ax  1a
0
1 + 2+ 1
1 + 3 + 3+ 1
1 + 4 + 6 + 4+ 1
 x  a   1x  3ax  3a x  1a
1 5 10
4
4
3
2 2
3
4
x

a

1
x

4
ax

6
a
x

4
a
x

1
a


3
3
2
2
3
10
5 1
Can you guess
the next row?
This is good for lower
powers but could get
very large. We will
introduce some
notation to help us and
generalize the
coefficients with a
formula based on what
was observed here.
1
1
1
1
1
1
4
5
1
2
3
6
1
3
4
10 10
1
1
5
1
This is called Pascal's Triangle and would give us the
coefficients for a binomial expansion of any power if we
extended it far enough.
Patterns observed
Consider the patterns formed by expanding (x + y)n.
•Powers on x and y add up to power on binomial
•x's increase in power as y's decrease in power from term to term.
(x + y)0 = 1
(x + y)1 = x + y
(x + y)2 = x2 + 2xy + y2
1. The exponents on x decrease
from n to 0 and the exponents
on y increase from 0 to n.
2. Each term is of degree n.
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y
Example: The 4th term of
5 is a term with x2y3.”
(x
+
y)
4
(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5
•Symmetry of coefficients (i.e. 2nd term and 2nd to last term have
same coefficients, 3rd & 3rd to last etc.) so once you've reached the
middle, you can copy by symmetry rather than compute coefficients.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
9
The coefficients of the binomial expansion are called binomial
coefficients. The coefficients have symmetry.
(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5
The first and last coefficients are 1.
The coefficients of the second and second to last terms
are equal to n.
Example 1: What are the last 2 terms of (x + y)10 ?
Since n = 10, the last two terms are 10xy9 + 1y10.
Example 2: What are the last 2 terms of (x + 2)10 ?
Since n = 10, the last two terms are 10x(2)9 + 1y10.
Use a calculator to calculate (2)9 = 510 then multiply it by 10
Your final answer should be 5100x + 1y10.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
10
Example 3: Use Pascal’s Triangle to expand (2a + b)4.
0th row
1
1
1
1
1
2
3
4
1st row
1
3
6
2nd row
1
3rd row
1
4
1
4th row
(2a + b)4 = 1(2a)4 + 4(2a)3b + 6(2a)2b2 + 4(2a)b3 + 1b4
= 1(16a4) + 4(8a3)b + 6(4a2b2) + 4(2a)b3 + b4
= 16a4 + 32a3b + 24a2b2 + 8ab3 + b4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
11
Extra: different form of questions you might see on test…
1. What is the second term in the
binomial expansion of this expression?
(x + 3)4
2. Find and simplify the fourth term in the expansion of
(3x2 + y3)4 .
3. (3x - 2y)4 =
[Hint: this is the same as (3x + -2y)4 so use a calculator to find the answer and write the whole
expansion for fourth power out for the answer.]