Chapter 8 Sec 5 The Binomial Theorem. 2 of 15 Pre Calculus Ch 8.5 Essential Question How do you find the expansion of the binomial (x + y) n ? Key Vocabulary:

Chapter 8 Sec 5

The Binomial Theorem

2 of 15

Pre Calculus Ch 8.5

Essential Question

How do you find the expansion of the binomial (x + y)n?

Key Vocabulary:Binomial Theorem

Pascal’s Triangle

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Pre Calculus Ch 8.5

Binomial CoefficientsTo begin this section, lets look at the expansion of (x + y)n for several values of n.

(x + y)0 = 1

(x + y)1 = x + y

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2y + 3xy2 + y3

(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5

1. In each there are n + 1 terms

2. In each x and y have symmetrical roles, power of x decrease by 1 and y increases by 1.

3. Sum of the powers equals n.

4. The coefficients increase then decrease symmetrically.

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Pre Calculus Ch 8.5

Binomial Theorem

Find each binomial coefficient

a. 8C2 b. c. 7C0 d.

a. 8C2

b.

c. 7C0

d.

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Pre Calculus Ch 8.5

¿7 ∙6 ∙53 ∙2 ∙1

¿7 ∙6 ∙53 ∙2 ∙1

Example 2Find each binomial coefficient

a. 7C3 b. 7C4 c. 12C1 d. 12C11

a. 7C3

b. 7C4

c. 12C1

d. 12C11

¿35

¿35

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Pre Calculus Ch 8.5

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Example 3Write the expansion of the expression (x + 1)3.

The binomial coefficients are

3C0 = 1, 3C1 = 3, 3C2 = 3, 3C3 = 1

1. In each there are n + 1 terms

2. In each x and y have symmetrical roles, power of x decrease by 1 and y increases by 1.

3. Sum of the powers equals n.

4. The coefficients increase then decrease symmetrically.

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Pre Calculus Ch 8.5

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Example 4Write the expansion of the expression (x – 1)3.

The binomial coefficients are

3C0 = 1, 3C1 = 3, 3C2 = 3, 3C3 = 1

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Pre Calculus Ch 8.5

Example 5Write the expansion of the expressiona. (2x – 3)4 b. (x – 2y)4

The binomial coefficients are

4C0 = 1, 4C1 = 4, 4C2 = 6, 4C3 = 4, 4C4 = 1

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Pre Calculus Ch 8.5

Example 6

Write the expansion of the expression (x2 + 4)3

The binomial coefficients are still

3C0 = 1, 3C1 = 3, 3C2 = 3, 3C3 = 1

(x2 + 4)3 = (1)(x2)3+ (3)(x2)2(4) + (3)(x2)(4)2 + (1)(4)3

= x6 + 12x4 + 48x2 + 64

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Pre Calculus Ch 8.5

Example 7Sometimes you will need to find a specific term in a binomial expansion. From the Binomial Theorem the (r + 1)th term is nCr xn – ryr

a. Find the sixth term of (a + 2b)8.

To find the sixth term, use n = 8 and r = 5 {the formula is for the (r + 1)st term, so r is one less than the number of the term you are looking for}

a. 8C5 a8 – 5(2b)5 = 56 ∙ a3 ∙ (2b)5 = 56(25)a3b5 = 1792a3b5

b. Find the coefficient of the term a6b5in the expansion of (2a – 5b)11.

b. 11C5 (2a)6(–5b)5 = (462)(64a6)(–3125b5)

= – 92,400,000a6b5

n = 11, r = 5, x = 2a, y = –5b

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Pre Calculus Ch 8.5

Pascal’s TrianglePascal gave us a convenient way to remember the pattern for binomial coefficients. 1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

First and last numbers are

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Pre Calculus Ch 8.5

Pascal’sLets put them together..

1

1x + 1y

1x2 + 2xy + 1y2

1x3 + 3x2y + 3xy2 + 1y3

1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5

1x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6

1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + 1y7

(x + y)0 =

(x + y)1 =

(x + y)2 =

(x + y)3 =

(x + y)4 =

(x + y)5 =

)6 =

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

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Pre Calculus Ch 8.5

Example 8Use the seventh row of Pascal’s Triangle to find the binomial coefficients.

1 7 21 35 35 21 7 1

8C0, 8C1, 8C2, 8C3, 8C4, 8C5, 8C6, 8C7, 8C8

Write the seventh row of Pascal’s Triangle

1 8 28 56 70 56 28 8 18C0 8C1 8C2 8C3 8C4 8C5 8C6 8C7 8C8

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Pre Calculus Ch 8.5

Essential Question

How do you find the expansion

of the binomial (x + y)n?

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Pre Calculus Ch 8.5

Daily Assignment

• Chapter 8 Section 5• Text Book

• Pg 624 – 625• # 1 – 33 Mode 4, 49 – 53

Odd; 57, 61, 69, 71, 79, 85

• Read Section 8.6• Show all work for credit.