1.

Warm-Up 4/7

E

Q&A on assignment.

1, 3, 7, 5 1, 3, 7, 5 2, 4, 7, 62, 4, 6

Rigor:You will learn how to use Pascal’s Triangle and the Binomial Theorem to expand a binomial.

Relevance: You will be able to use these skills in future

math courses.

10-5 Binomial Theorem

nCr = n (math) (>) (>) (>) (3) r

3C2 = 3

5C1 = 5

4C0 = 1

7C7 = 16C4 = 15

8C2 = 28

Pascal’s Triangle

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, …

Example 1a: Use Pascal’s Triangle to expand the binomial.

(𝑎+𝑏 )7

Step 1 Write series omitting coefficients.

Step 2 Use a row of numbers in Pascal’s Triangle as the coefficients.

(𝑎+𝑏 )7

(𝑎+𝑏 )7

Example 1b: Use Pascal’s Triangle to expand the binomial.

(3 𝑥+2 )4

Step 1 Write series omitting coefficients. Let

Step 2 Use a row of numbers in Pascal’s Triangle as the coefficients.

Example 2: Use Pascal’s Triangle to expand the binomial.

(𝑥− 4 𝑦 )5

Step 1 Write series omitting coefficients. Let

Step 2 Use a row of numbers in Pascal’s 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

∑𝑟=4

7

7𝑐4𝑎7−4 (𝑏)4

7𝑐4𝑎7 − 4(𝑏)4

3 5𝑎3𝑏4

35

Example 4: Find the coefficient of in the expansion of .

n = 9 a = 4x b = – 3y r = ?

∑𝑟=?

9

9𝑐? (4 𝑥 )9−? (−3 𝑦)?

9𝑐2 ¿(36)¿

(36)(16384)(9)𝑥7 𝑦 2

r = 2

5,308,416 𝑥7 𝑦2

∑𝑟=2

9

9𝑐2(4 𝑥)9 − 2(−3 𝑦)2

5,308,416

Example 6: Use the Binomial Theorem to expand the binomial.

(3 𝑥− 𝑦 )4

n = 4 a = 3x b = – y

∑𝑟=0

4

4𝑐𝑟 (3 𝑥)4− 𝑟(− 𝑦 )𝑟

4𝑐0(3 𝑥)4 −0(− 𝑦 )0+4𝑐1(3 𝑥)4 −1(− 𝑦)1+4𝑐2(3 𝑥)4 −2(− 𝑦 )2+4𝑐3(3 𝑥)4 −3(−𝑦 )3+4𝑐4 (3 𝑥)4 − 4(−𝑦 )4

(1) +( 4 ) (3 𝑥 )3(− 𝑦 )+(6) (3 𝑥 )2(𝑦2)+(4 ) (3𝑥 )1(− 𝑦3)+(1)(1)(𝑦4)

81 𝑥4+( 4 )(27 𝑥¿¿3)(− 𝑦 )¿+(6)(9𝑥¿¿2)(𝑦2)¿+(4 )(3 𝑥)(− 𝑦3)+(𝑦 4)

(3 𝑥− 𝑦 )4=¿81 𝑥4−108 𝑥3 𝑦+54 𝑥2 𝑦2−12 𝑥 𝑦 3+𝑦4

Example 7: Represent the expansion using sigma notation.

(5 𝑥−7 𝑦 )20

n = 20 a = 5x b = – 7y

∑𝑟=0

20

20𝑐𝑟 (5 𝑥)20 −𝑟 (−7 𝑦 )𝑟(5 𝑥−7 𝑦 )20=¿

∑𝑟=0

20

20𝑐𝑟 (5 𝑥)20 −𝑟 (−7 𝑦 )𝑟(5 𝑥−7 𝑦 )20=¿

∑𝑟=0

20

(20𝑟 )(5 𝑥)20− 𝑟(− 7 𝑦 )𝑟(5 𝑥−7 𝑦 )20=¿

or

√−1math!

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.

(2𝑝+𝑞2 )5

n = 5 a = 2p b = q2

∑𝑟=0

5

5𝑐𝑟 (2𝑝)5 −𝑟 (𝑞2)𝑟

5𝑐0(2𝑝)5− 0(𝑞2)0+5𝑐1(2𝑝)5 − 1(𝑞2)1+5𝑐2(2𝑝)5 − 2(𝑞2)2+5𝑐3(2𝑝 )5−3(𝑞2)3+5𝑐4 (2𝑝)5− 4(𝑞2)4+5𝑐5(2𝑝)5 − 5(𝑞2)5

(1) +(5 ) (2𝑝 )4 (𝑞2)+(10) (2𝑝 )3(𝑞4)+(10) (2𝑝 )2(𝑞6) +(1)(1)(𝑞10)+(5) (2𝑝 )1(𝑞8)

(1) +(5 )(16𝑝¿¿ 4)(𝑞2)¿+(10)(8𝑝¿¿ 3)(𝑞4)¿+(10)(4𝑝¿¿2)(𝑞6)¿ +(1)(1)(𝑞10)+(5)(2𝑝)(𝑞8)

32𝑝5+80𝑝4𝑞2+80𝑝3𝑞4+40𝑝2𝑞6 +𝑞10+10𝑝𝑞8=