2.4Use the Binomial Theorem

2.1-2.5 Test: Friday

Think about this…

Expand (x + y)12

Would you want to multiply (x +y) 12 times?!?!?!

Vocabulary

Binomial Theorem and Pascal’s Triangle The numbers in Pascal’s triangle can be

used to find the coefficients in binomial expansions (a + b)n where n is a positive integer.

Vocabulary

Binomial Expansion(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2

(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3

(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4

Vocabulary

Binominal Expansion(a + b)3

1a3 + 3a2b + 3ab2 + 1b3

*as the a exponents decrease, the b exponents increase

Where do the coefficients (1, 3, 3, 1) come from?

Vocabulary

Pascal’s Triangle 1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

n = 0 (0th row)

n = 1 (1st row)

n = 2 (2nd row)

n = 3 (3rd row)

n = 4 (4th row)

The first and last numbers in each row are 1. Beginning with the 2nd row, every other number is formed by adding the two numbers immediately above the number

Example:

Use the forth row of Pascal’s triangle to find the numbers in the fifth row of Pascal’s triangle.

1 4 6 4 1

1 5 10 10 5 1

Example:

Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x + 2)3

Binomial Theorem: (a + b)3

= a3 + a2b + ab2 + b3

Pascal’s Triangle: row 3

1 3 3 1

Together: 1a3 + 3a2b + 3ab2 + 1b3

Example Continued:

(x + 2)3

a b

1a3 + 3a2b + 3ab2 + 1b3

1(x)3 + 3(x)2(2) + 3(x)(2)2 + 1(2)3

X3 + 6x2 + 12x + 8

You Try:

Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x + 1)4

Solution:a = x, b = 1

1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4

x4 + 4x3 + 6x2 + 4x + 1

Example:

Use the Binomial Theorem and Pascal’s Triangle to write the binomial expansion of (x – 3)4

watch out for the negative!

1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4

1(x)4 + 4(x)3(-3) + 6(x)2(-3)2 + 4(x)(-3)3 + 1(-3)4

x4 – 12x3 + 54x2 – 108x + 81

a b

Homework:

p. 75 # 1-13odd Due tomorrow