Download PASCAL’S TRIANGLE & BINOMIAL EXPANSIONS

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PASCAL’S TRIANGLE & BINOMIAL EXPANSIONS
PART A ~ WHO IS PASCAL & WHAT IS HIS TRIANGLE?!?
The array of numbers shown below is called Pascal’s triangle in honour of French
mathematician, Blaise Pascal (1623–1662). Each row is generated by calculating the
sum of pairs of consecutive terms in the previous row.
1
1
1
1
1
1
1
6
1
2
3
4
5
Row 0
1
3
6
10
15
Row 1
1
4
10
20
Row 2
Row 3
1
5
15
Row 4
1
6
Row 5
1
Row 6
Row 7
Row 8
Complete rows 7 & 8.
PART B ~ BINOMIAL EXPANSIONS (𝒂 + 𝒃)𝒏
Expand each of the following:
a)
(𝑎 + 𝑏)2 =
b)
(𝑎 + 𝑏)3 =
c)
(𝑎 + 𝑏)4 =
Examine the coefficients in the above examples. Is there a connection between the
coefficients and Pascal’s triangle?
Examine the literal coefficients in the above examples. Is there a pattern with the
exponents of 𝑎 and 𝑏?
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In general, when expanding any binomial (𝑎 + 𝑏)𝑛 :
 the coefficients of each term follow Pascal’s Triangle (n = row #)
 the pattern of exponents on a and b follows:
(a  b)n  a n  a n1b  a n2b2  a n3b 3  a n4 b 4 ...  b n
Note: If the exponent is n, there will be n + 1 terms in the expansion.
PART
C ~ EXAMPLES

Ex 
Expand each of the following binomials:
a)
(𝑥 + 𝑦)5 =
b)
(x − y)5 =
c)
(2x − 1)6 =
d)
(3𝑥 2 − 2y)4 =
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Determine the value of k in each term from the binomial expansion of (𝑎 + 𝑏)11 :
Ex 
a)
Ex 
462𝑎6 𝑏𝑘
b)
330𝑎𝑘 𝑏 4
Determine the number of terms in the expansion of each of the following:
a)
(2𝑎 + 3𝑏)12
b)
(2𝑥 − 3𝑦)27
Ex 
Write and simplify the fourth term in the expansion of (3𝑥 2 − 4𝑦)8 .
Ex 
Write 1 + 10𝑥 2 + 40𝑥 4 + 80𝑥 6 + 80𝑥 8 + 32𝑥 10 in the form (𝑎 + 𝑏)𝑛 .
HOMEWORK: p.466 #1, 2, 4bdf, 5ace