Example 9. Expand (x + 2)3.
Solution. The coefficients are 1 3 3 1. x is "a", and 2 is "b".
(x + 2)3 = x3 + 3x²· 2 + 3x· 2² + 23
= x3 + 6x² + 12x + 8
Problem 1. In the expansion of (a + b)n, each term has the form
an − kbk,
where k successively takes on the value 0, 1, 2, . . ., n.
is the symbol for the binomial coefficient.
The binomial theorem is the statement that
=?
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The combinatorial number, nCk.
Problem 2. Use factorials to write the general term in the
expansion of (a + b)n.
n!
a n − k bk
(n − k)! k!
Problem 3.
a) Calculate the coefficient of a4b6 in the expansion of (a + b)10.
C6 = 10C4 = 210
10
b) The coefficient of which other term is the same? a6b4
c) In the expansion of (a + b)n, the coefficient of an − kbk is the
same as
c) the coefficient of which other term? akbn − k
Problem 4. Calculate the coefficient of
a) x17y3 in the expansion of (x + y)20. 1140
b) x3y17 in the expansion of (x + y)20. 1140
c) x3y17 in the expansion of (x − y)20. −1140
d) x2y18 in the expansion of (x − y)20. 190
e) x5y5 in the expansion of (x − y)10. −252
f) x10 in the expansion of (x − 1)15. −3003
Problem 5. Write the first four terms of (x + h)n. Do not use
factorials.
(x + h)n = xn + nxn−1h +
n(n − 1)
1· 2
xn−2h2 +
n(n − 1)(n − 2)
1· 2· 3
xn−3h3
Problem 6. Compute the first four terms of each of the
following.
a) (a + b)15 a15 + 15a14b + 105a13b² + 455a12b3
b) (x − 1)20 x20 − 20x19 + 190x18 − 1140x17
Problem 7. Consider the expansion of (x + b)30.
a) What is the exponent of b in the 1st term? 0
b) What is the exponent of b in the 3rd term? 2
c) In the 25th term? 24
d) In the kth term? k − 1
e) Write the fourth term, with its coefficient. 4,060x27b3
Problem 8. Calculate each of the following.
a) The third term of (a + b)11. 55a9b²
b) The fifth term of (x − y)7. 35x3y4
c) The tenth term of (x − 1)12. −220x3
1 18
−14
d) The fifteenth term of (1 + ) . 3060x
x
1 10
4
e) The fourth term of (x − ) . −120x
x
Problem 9. Use Pascal's triangle to expand the following.
a) (a + b)3 = a3 + 3a²b + 3ab² + b3
b) (a − b)3 = a3 − 3a²b + 3ab² − b3
c) (x + y)4 = x4 + 4x3y + 6x²y² + 4xy3 + y4
d) (x − y)4 = x4 − 4x3y + 6x²y² − 4xy3 + y4
e) (x − 1)5 = x5 − 5x4 + 10x3 − 10x² + 5x − 1
f) (x + 2)5 = x5 + 10x4 + 40x3 + 80x² + 80x + 32
g) (2x − 1)3 = 8x3 − 12x² + 6x − 1
h) (1 − xy)7
= 1 − 7xy + 21x²y² − 35x3y3 + 35x4y4 − 21x5y5 + 7x6y6 − x7y7
In the following Topic we will give a proof of the binomial
theorem.