Warm up • 1. Write the expression in expanded form, then find the sum. • 2. Express the series using sigma notation. 5 1 ) 5 6 ( n n 13 11 9 7 5
Jan 04, 2016
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