Warm up 1. Write the expression in expanded form, then find the sum. 2. Express the series using sigma notation.

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 tetrahedra l 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 Theorem Solutions 1.(x + 2) 2 = x 2 + 2(2)x = x 2 + 4x (2x + 3) 2 = (2x) 2 + 2(3)(2x) = 4x x (x – 3) 3 = (x – 3)(x – 3) 2 = (x – 3)(x 2 – 2(3)x ) = (x – 3)(x 2 – 6x + 9) = x(x 2 – 6x + 9) – 3(x 2 – 6x + 9) = x 3 – 6x 2 + 9x – 3x x – 27 = x 3 – 9x x – 27 4.(a + b) 4 = (a + b) 2 (a + b) 2 = (a 2 + 2ab + b 2 )(a 2 + 2ab + b 2 ) = a 2 (a 2 + 2ab + b 2 ) + 2ab(a 2 + 2ab + b 2 ) + b 2 (a 2 + 2ab + b 2 ) = a 4 + 2a 3 b + a 2 b 2 + 2a 3 b + 4a 2 b 2 + 2ab 3 + a 2 b 2 + 2ab 3 + b 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4

THAT is a LOT of work! Isn’t there an easier way?

The Binomial Theorem

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 a 5 + 5a 4 b + 10a 3 b a 2 b 3 + 5ab 4 + b 5. The exponents for a begin with 5 and decrease. 1a 5 b 0 + 5a 4 b a 3 b a 2 b 3 + 5a 1 b 4 + 1a 0 b 5 The exponents for b begin with 0 and increase. Row 5

The Binomial Theorem The expansion of has n+1 terms The 1 st 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.

Expand

Expanding with coefficients (2x – y) 4 =16x 4 + 4(8x 3 )(-y) + 6(4x 2 )(y 2 ) + 4(2x)(-y 3 ) + y 4 = 16x 4 – 32x 3 y + 24x 2 y 2 – 8xy 3 + y 4

Expand

Find the fourth term of (2x-3y) 6