Download 2.4 Use the Binomial Theorem

2.4
Use the Binomial Theorem
2.1-2.5 Test: Friday
Think about this…
 Expand (x + y)12
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
n = 0 (0th row)
1 1
1
1
1
2
3
4
n = 1 (1st row)
1
3
6
n = 2 (2nd row)
1
4
n = 3 (3rd row)
1
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:
Binomial
(a + b)3Theorem
 UseTheorem:
the Binomial
and
2 + b3
Triangle
to write the binomial
= a3 Pascal’s
+ a2b + ab
expansion of (x + 2)3
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!
a
b
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
Homework:
 p. 75 # 1-13odd
 Due tomorrow