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6.3
Pascal’s Triangle and Expanding
Binomial Powers
Pascal’s triangle Pascal’s triangle
• a triangular
arrangement of
numbers with 1 in the
first row, and 1 and 1 in
the second row
• Each number in the
succeeding rows is
the sum of the two
numbers above it in the
preceding row.
Investigate Tools
Optional
How can you use patterns to expand a power of a binomial? • Computer Algebra
System (CAS)
1.
Te c h n o l o g y Tip
a) a b You can use the CAS
engine of a TI-Nspire™
b) a b CAS graphing
calculator to expand a
c) a b power of a binomial.
• Open a new
d) a b calculator page.
• Press b, select
2. Reflect
3:Algebra, and then
select 3:Expand.
• Enter the power
3. Reflect
of a binomial. For
example, type
(a + b)3 and press ·.
4. a b
The expansion will be
displayed.
6.3 Pascal’s Triangle and Expanding Binomial Powers • MHR 373
Example 1
Patterns in Pascal’s Triangle
a)
b)
Solution
a)
b)
Example 2
Position of Terms in Pascal’s Triangle
tn r
n
r
n r t
n r t t
n r t t t
n r t t t t
n r t t t t t
n r t t t t t t
n r t t t t t t t
tn r
tn r
tn r
t t tn r
374 MHR • Functions 11 • Chapter 6
Solution
t t t
t
t t
t t t
t t t t
t t t t t
t t t t t t
t t t t t t t
Example 3
Relate Other Patterns to Pascal’s Triangle
a)
b)
c)
Solution
a)
b)
c)
Term
Number, n Term, f(n) First
Differences Second
1 1 Differences
2
2 3 1
3
3 6 1
4
4 10
a
f n an bn c.
f n n bn c
6.3 Pascal’s Triangle and Expanding Binomial Powers • MHR 375
b c
b c
b c
b c
n f n n n
f f n f n n
Con ne ct io ns
n
If you take Mathematics
a b n
of Data Management in
grade 12, you will see Value of n (a + b)n
how Pascal’s triangle 0 (a + b)0 = 1
and expansions of
1 (a + b)1 = a + b
(a + b)n are connected to
probability. 2 (a + b)2 = a 2 + 2ab + b 2
3 (a + b)3 = a 3 + 3a 2b + 3ab 2 + b 3
4 (a + b)4 = a 4 + 4a 3b + 6a 2b 2 + 4ab 3 + b 4
a b
a b
Example 4
Expand a Power of a Binomial
y
a) a b b) m n c) x d) y
Solution
a)
a b
a b ab ab ab ab ab
ab ab ab
a ab ab ab ab ab ab b
376 MHR • Functions 11 • Chapter 6
b) a m
b n
m n m n m n m n m n
m n m n
m mn mn mn mn n
c) a x
b
x x x x x
x x x
x x x x x x
y
d) a
b y
y y y y y
y y y y y
y
y
y y y
y y
Key Concepts
tn r
tn r
tn r
n r
n r r n
n
a b n
Communicate Your Understanding
C1 t t
C2
C3
C4 a b
6.3 Pascal’s Triangle and Expanding Binomial Powers • MHR 377
A Practise
For help with questions 1 and 2, refer to 7. Reasoning and Proving
Example 1. Representing Selecting Tools
1. 1 Problem Solving
1 1
Connecting Reflecting
1 2 1 k
1 3 3 1 Communicating
x y
1 4 6 4 1
1 5 10 10 5 1 a) ky b) xky
c) x yk d) kx y
B Connect and Apply
8.
a) b)
c) d)
2. 9.
tn r
a) b) a) t b) t c) t d) t
c) d) n
10.
For help with questions 3 and 4, refer to
Example 2. a) b)
3. 15 6
tn r
36 9 1 7
a) t t b) t t
c) t t d) ta ta 56
b b
4.
c) d)
tn r
10 45 35
a) t b) t c) t d) t x
28 56
For help with questions 5 to 7, refer to
Example 3.
5.
11. Chapter Problem
a) x b) y c) t 1
d) m e) x y f) a 1 1
6. 1 2 1
a) a b) x 1 3 3 1
n
c) t d) b a
378 MHR • Functions 11 • Chapter 6
12. 17.
Hint:
C Extend
13. Reasoning and Proving
Representing Selecting Tools
Problem Solving
Connecting Reflecting
Communicating
18. Math Contest
n
14. x
x n
15.
A B C D
19. Math Contest
x
x
16. A B C D
20. Math Contest
A B C D
6.3 Pascal’s Triangle and Expanding Binomial Powers • MHR 379
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