Rigor:You will learn how to use Pascalβs Triangle and the Binomial Theorem to expand a binomial.
Relevance: You will be able to use these skills in future
math courses.
Binomial Theorem: The binomial expansion of (a + b)n for any positive integer n is:
1st term r = 0, 2nd term r = 1, 3rd term r = 2, 4th term r = 3, β¦
Example 1a: Use Pascalβs Triangle to expand the binomial.
(π+π )7
Step 1 Write series omitting coefficients.
Step 2 Use a row of numbers in Pascalβs Triangle as the coefficients.
(π+π )7
(π+π )7
Example 1b: Use Pascalβs Triangle to expand the binomial.
(3 π₯+2 )4
Step 1 Write series omitting coefficients. Let
Step 2 Use a row of numbers in Pascalβs Triangle as the coefficients.
Example 2: Use Pascalβs Triangle to expand the binomial.
(π₯β 4 π¦ )5
Step 1 Write series omitting coefficients. Let
Step 2 Use a row of numbers in Pascalβs Triangle as the coefficients.
Example 3: Find the coefficient of the 5th term in the expansion of .
n = 7 a = a b = b r = 4
βπ=4
7
7π4π7β4 (π)4
7π4π7 β 4(π)4
3 5π3π4
35
Example 4: Find the coefficient of in the expansion of .
n = 9 a = 4x b = β 3y r = ?
βπ=?
9
9π? (4 π₯ )9β? (β3 π¦)?
9π2 ΒΏ(36)ΒΏ
(36)(16384)(9)π₯7 π¦ 2
r = 2
5,308,416 π₯7 π¦2
βπ=2
9
9π2(4 π₯)9 β 2(β3 π¦)2
5,308,416
Example 6: Use the Binomial Theorem to expand the binomial.
(3 π₯β π¦ )4
n = 4 a = 3x b = β y
βπ=0
4
4ππ (3 π₯)4β π(β π¦ )π
4π0(3 π₯)4 β0(β π¦ )0+4π1(3 π₯)4 β1(β π¦)1+4π2(3 π₯)4 β2(β π¦ )2+4π3(3 π₯)4 β3(βπ¦ )3+4π4 (3 π₯)4 β 4(βπ¦ )4
(1) +( 4 ) (3 π₯ )3(β π¦ )+(6) (3 π₯ )2(π¦2)+(4 ) (3π₯ )1(β π¦3)+(1)(1)(π¦4)
81 π₯4+( 4 )(27 π₯ΒΏΒΏ3)(β π¦ )ΒΏ+(6)(9π₯ΒΏΒΏ2)(π¦2)ΒΏ+(4 )(3 π₯)(β π¦3)+(π¦ 4)
(3 π₯β π¦ )4=ΒΏ81 π₯4β108 π₯3 π¦+54 π₯2 π¦2β12 π₯ π¦ 3+π¦4
Example 7: Represent the expansion using sigma notation.
(5 π₯β7 π¦ )20
n = 20 a = 5x b = β 7y
βπ=0
20
20ππ (5 π₯)20 βπ (β7 π¦ )π(5 π₯β7 π¦ )20=ΒΏ
βπ=0
20
20ππ (5 π₯)20 βπ (β7 π¦ )π(5 π₯β7 π¦ )20=ΒΏ
βπ=0
20
(20π )(5 π₯)20β π(β 7 π¦ )π(5 π₯β7 π¦ )20=ΒΏ
or
ββ1math!
10-5 Assignment: TX p633, 8-24 EOE & 36-44 EOEAssignment: Review 2-5 & 2-6 1-18 allTest on 2-5, 2-6, 6-4 & 10-5 Thursday 4/10
Example 6b: Use the Binomial Theorem to expand the binomial.
(2π+π2 )5
n = 5 a = 2p b = q2
βπ=0
5
5ππ (2π)5 βπ (π2)π
5π0(2π)5β 0(π2)0+5π1(2π)5 β 1(π2)1+5π2(2π)5 β 2(π2)2+5π3(2π )5β3(π2)3+5π4 (2π)5β 4(π2)4+5π5(2π)5 β 5(π2)5
(1) +(5 ) (2π )4 (π2)+(10) (2π )3(π4)+(10) (2π )2(π6) +(1)(1)(π10)+(5) (2π )1(π8)
(1) +(5 )(16πΒΏΒΏ 4)(π2)ΒΏ+(10)(8πΒΏΒΏ 3)(π4)ΒΏ+(10)(4πΒΏΒΏ2)(π6)ΒΏ +(1)(1)(π10)+(5)(2π)(π8)
32π5+80π4π2+80π3π4+40π2π6 +π10+10ππ8=