**Unformatted text preview: **increases by one as we move from left to right, we quickly obtain
(3x − y )4 = (1)(3x)4 + (4)(3x)3 (−y ) + (6)(3x)2 (−y )2 + 4(3x)(−y )3 + 1(−y )4
= 81x4 − 108x3 y + 54x2 y 2 − 12xy 3 + y 4
We would like to stress that Pascal’s Triangle is a very quick method to expand an entire binomial.
If only a term (or two or three) is required, then the Binomial Theorem is deﬁnitely the way to go. 590 9.4.1 Sequences and the Binomial Theorem Exercises 1. Simplify the following expressions.
(d) (b) (c) 7!
23 3! (g) 8
3 (n + 1)!
, n ≥ 0.
n! (h) 117
0 (f) 10!
7! 9!
4!3!2! (e) (a) (3!)2 (k − 1)!
, k ≥ 1.
(k + 2)! (i) n
,n≥2
n−2 2. Use Pascal’s Triangle to expand the following.
(a) (x + 2)5 (b) (2x − 1)4 1
3x (c) + y2 3 (d) x − x−1 4 3. Use Pascal’s Triangle to simplify the following powers of complex numbers.
(a) (1 + 2i)4
√
(b) −1 + i 3 √
3 √
2
2
−i
2
2 (c) 4 4. Use the Binomial Theorem to ﬁnd the indicated term in the following expansions.
(a) The ter...

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