Warm up

• 1. Write the expression in expanded form, then find the sum.

• 2. Express the series using sigma notation.

5

1

)56(n

n

1311975

Lesson 12-6 The Binomial Theorem

Objective: To use the Pascal’s Triangle to expand binomials

Introducing: Pascal’s Triangle

What

patterns do you see?Row 5

Row 6

• counting numbers

• triangular numbers

• tetrahedral numbers

• The sum of each row is a power of 2.

• rows are powers of eleven

Serpinski’s Triangle

The Binomial Theorem

Strategy only: how do we expand these?

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

3. (x – 3)3 4. (a + b)4

The Binomial TheoremSolutions

1. (x + 2)2 = x2 + 2(2)x + 22 = x2 + 4x + 4

2. (2x + 3)2 = (2x)2 + 2(3)(2x) + 32 = 4x2 + 12x + 9

3. (x – 3)3 = (x – 3)(x – 3)2 = (x – 3)(x2 – 2(3)x + 32) =(x – 3)(x2 – 6x + 9) = x(x2 – 6x + 9) – 3(x2 – 6x + 9) =x3 – 6x2 + 9x – 3x2 + 18x – 27 = x3 – 9x2 + 27x – 27

4. (a + b)4 = (a + b)2(a + b)2 = (a2 + 2ab + b2)(a2 + 2ab + b2) =a2(a2 + 2ab + b2) + 2ab(a2 + 2ab + b2) + b2(a2 + 2ab + b2) =a4 + 2a3b + a2b2 + 2a3b + 4a2b2 + 2ab3 + a2b2 + 2ab3 + b4 =a4 + 4a3b + 6a2b2 + 4ab3 + b4

THAT is a LOT of work!

Isn’t there an easier way?

The Binomial Theorem

32233

222

10011

0

1331)(

121)(

11)(

1)(

yxyyxxyx

yxyxyx

yxyxyxyx

yx

Use Pascal’s Triangle to expand (a + b)5.The Binomial Theorem

Use the row that has 5 as its second number.

In its simplest form, the expansion is a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5.

The exponents for a begin with 5 and decrease.

1a5b0 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1a0b5

The exponents for b begin with 0 and increase.

Row 5

The Binomial Theorem

• The expansion of has n+1 terms• The 1st term is and the last term is• The x exponent decreases by 1 each term and

the y exponent increases by 1 each term• The degree of each term is n• The coefficients are symmetric.

nyx )( nx ny

Expand8)( yx

87625344

3526788

8285670

56288)(

yxyyxyxyx

yxyxyxxyx

Expanding with coefficients

(2x – y)4 =16x4 + 4(8x3)(-y) + 6(4x2)(y2) + 4(2x)(-y3) + y4

= 16x4 – 32x3y + 24x2y2 – 8xy3 + y4

Expand7)5( yx

• Find the fourth term of (2x-3y)6