Multiply out to find the coefficient of the x2 y3 term in the expansion of (x+y)5
Solution = 10
Binomial Theorem; Pascal's Triangle
sec 9­5
(a+b)1 = 1a+1b
(a+b)2 = 1a2 +2ab+b 2
(a+b)3 = 1a3 +3a2 b+3ab2 +1b3
(a+b)4 = 1a4 +4a3 b+6a2 b2 +4ab3 +1b4
(a+b)5 = 1a5 +5a4 b+10a3 b2 +10a2 b3 +5ab4 +1b5
What patterns do we see?
•The first term of (a+b) n is always 1an b0
• In successive terms the exponents of a decrease by 1 and the exponents of b increase by one.
• the sum of the two exponents in any term is equal to n.
• The coefficients of the terms form Pascal's Triangle.
1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
What would the 6th row be?
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
Nov 11­7:46 AM
Example:
Use the Binomial Theorem to expand and simplify the expression.
1) (x + 1) 6
use 6th row of Pascals Triangle...
1 6 15 20 15 6 1
1x6(1)0 + 6(x)5(1)1 + 15(x)4(1)2 + 20(x)3(1)3 + 15(x)2(1)4 + 6(x)1(1)5 + 1(x)0(1)6
x6 + 6x5 + 15x4 + 20x3 + 15x2 + 6x + 1
2) (2x - 5y)5
Use 5th row of
Pascal's Triangle...
Alternate signs in
1 5 10 10 5 1
expansion with
differences.
1(2x)5(5y)0 - 5(2x)4(5y)1 + 10(2x) 3(5y)2 - 10(2x)2(5y)3 + 5(2x) 1(5y)4 - 1(2x) 0(5y)5
Simplify and multiply...
32x5 - 400x4y + 2000x3y2 - 5000x2y3 + 6250xy4 - 3125y5
Example:
Expand the binomial by using Pascal's Triangle to
determine the coefficients.
These directions in your homework mean to
do the same thing we did above.
Oct 22­1:39 PM
1
3) (y - 2)5
4) (x + 2y) 4
Oct 24­12:25 PM
2