Download Lesson 6: Pascal`s Triangle and Binomial Expansion

Lesson 6: Pascal’s Triangle and Binomial Expansion
Recall: The Fibonacci Sequence: 𝑡1 = 1, 𝑡2 = 1, 𝑡𝑛 = 𝑡𝑛−1 + 𝑡𝑛−2 , 𝑛 ∈ 𝑁, 𝑛 > 2
Part A – Pascal’s 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
Row 0
1
1
1
1
1
2
3
4
Row 1
1
3
6
Row 2
1
4
Row 3
1
Row 4
Row 5
Part B – Binomial Expansion (𝒂 + 𝒃)
 Expand the following:
a) (𝑎 + 𝑏)2 =
𝒏
Complete rows 5 & 6
Row 6
b) (𝑎 + 𝑏)3 =
c) (𝑎 + 𝑏)4 =
Notice the pattern of exponents on a and b above.
𝒂𝒏 , 𝒂𝒏−𝟏 𝒃, 𝒂𝒏−𝟐 𝒃𝟐 , 𝒂𝒏−𝟑 𝒃𝟑 , 𝒂𝒏−𝟒 𝒃𝟒 , ⋯ ⋯ , 𝒂𝟐 𝒃𝒏−𝟐 , 𝒂𝟐 𝒃𝒏−𝟏 , 𝒃𝒏
Look at the coefficients in the above examples. Is there a connection between
them and Pascal’s triangle?
∴ (𝑎 + 𝑏)𝑛 =? 𝑎𝑛 +? 𝑎𝑛−1 𝑏+? 𝑎𝑛−2 𝑏 2 +? 𝑎𝑛−3 𝑏 3 +? 𝑎𝑛−4 𝑏 4 ⋯ ⋯ +? 𝑎2 𝑏 𝑛−2 +? 𝑎2 𝑏 𝑛−1 +? 𝑏 𝑛
where ? is from the nth row of Pascal’s Triangle.
Note: For (𝑎 + 𝑏)𝑛 the exponents for each term in the expansion must add to n,
and there will be 𝑛 + 1 terms.
Part C – More Examples
Example 1: Expand the following using the patterns you just observed.
a) (𝑥 + 𝑦)5
a
b
n = ____ → ____ terms and exponents for each term add to ____
Pascal’s Triangle row _____ → ______________________________
(𝑥 + 𝑦)5 =
b) (𝑥 + 𝑦)8
a
b
n = ____ → ____ terms and exponents for each term add to ____
Pascal’s Triangle row _____ → ______________________________
(𝑥 + 𝑦)8 =
c) (𝑥 − 𝑦)4
a
b
n = ____ → ____ terms and exponents for each term add to ____
Pascal’s Triangle row _____ → ______________________________
(𝑥 − 𝑦)4 =
d) (2𝑥 − 1)4
a
b
n = ____ → ____ terms and exponents for each term add to ____
Pascal’s Triangle row _____ → ______________________________
(2𝑥 − 1)4 =
e) (3𝑥 − 2𝑦)5 n = ____ → ____ terms and exponents for each term add to ____
a
b
Pascal’s Triangle row _____ → ______________________________
(3𝑥 − 2𝑦)5 =
Note: For the binomial expansions where b is negative, the terms will alternate
positive and negative
Example 2: Determine the value of k in each from the binomial expansion of a  b 11.
a) 462𝑎6 𝑏 𝑘
b) 330𝑎𝑘 𝑏 4
Example 3: Determine the number of terms in the expansion if,
a) (2𝑎 + 3𝑏)12
b) (4𝑥 − 3𝑦)27
Example 4: Factoring Using the Binomial Theorem:
Write 64p3 + 240p2q + 300pq2 + 125q3 in the form (𝑎 + 𝑏)𝑛 .
Example 5: Factoring Using the Binomial Theorem:
Write 1 + 10𝑥 2 + 40𝑥 4 + 80𝑥 6 + 80𝑥 8 + 32𝑥 10 in the form (𝑎 + 𝑏)𝑛 .