1. Section 13.5 Expanding Binomials
Example 1.1. Expand each binomial...
(1) (a + b)2 =
(2) (a + b)3 =
(3) (a + b)4 =
2. Pascal’s Triangle
n=0
n=1
n=2
n=3
n=4
n=5
1
1
1
1
1
1
2
1
3
4
1
Example 2.1. Find (2x − 3)3 .
1
Section 13.5 Expanding Binomials
2
3. n Choose j
n
Definition 3.1. The Binary Coefficient or “n choose j” =
=n Cj = C(n, j) =
j
12
Example 3.1. Evaluate
10
12
Example 3.2. Evaluate
2
Example 3.3. Evaluate
(1)
(2)
(3)
(4)
n
0
n
n
n
1
n
n−1
Section 13.5 Expanding Binomials
4. The Binomial Theorem
Theorem 4.1. (a + b)n
n n 0
n n−1 1
n n−2 2
n
n 0 n
1 n−1
=
a b +
a b +
a b + ··· +
ab
+
ab
0
1
2
n−1
n
=
n X
n
j=0
j
an−j bj
Theorem 4.2. The form of each term in the expansion of (a + b)n is
Example 4.1. Find (2x − 3)5 .
3
Section 13.5 Expanding Binomials
4
Example 4.2. Find the coefficient of x5 in the expansion of (4 − x)12
Example 4.3. Find the coefficient of x6 in the expansion of (4x2 + y)12
Example 4.4. Find the term that does not contain any y (or contains y 0 ) in the
expansion of (xy + y −2 )12
Example 4.5. Find the fifth term in the expansion of (3a2 −
decreasing order of powers of the first term.
√
b)9 if arranged in