IB Mathematics SL & HL
exercise set 1-1-30v1
Binomial Expansions & Binomial Theorem
binomial theorem – in formula booklet:
 a  b
n
n
 a n    a n1b 
1
 n
   a nr br 
r
 bn
binomial theorem – using sigma notation:
 a  b
n
n
n
    a nr br
r 0  r 
Exercises
[ answers on next page ]
Part I - questions 1-5 ▪ no calculator ▪
1.
Perform each of the following expansions.
(a)
2.
4
 x  1
(b)
4
(c)
 x  2
4
Expand:
(a)
3.
 x  1
 a  b
4
(b)
 a  b
4
(c)
 3a  b 
4
(d)
 3a  2b 
4
Expand each of the following:
(a)
x
2
 1
5
(b)
x
3
2 

(c)  x 2  4 
x 

 2x 
5
4.
Find the term containing x 3 in the expansion of  3x  1 .
5.
2

Find the constant term in the expansion of  a   .
a

5
4
6
Part II - questions 6-10 ▪ calculator allowed ▪
6.
Consider the expansion of  x  2  .
11
(a) Write down the number of terms in this expansion.
(b) Find the term containing x 2 .
7.
Find the term containing x 4 in the expansion of  2 x  3 .
8.
The second term in the expansion of  x  p  is 150x 2 . Find the possible values of p.
9.
Find the coefficient of x 4 in the expansion of  x  1  2 x  1 .
7
4
2
4
6
c 

10. Consider the expansion of x  2 x3  2  . The constant term is 4860. Find the possible
x 

values of c.
2
© InThinking – IB Maths HL & SL
1
IB Mathematics SL & HL
exercise set 1-1-30v1
Binomial Expansions & Binomial Theorem
ANSWERS
1.
(a) x 4  4 x3  6 x 2  4 x  1
2.
(a) a 4  4a 3b  6a 2b 2  4ab3  b 4
(b) a 4  4a 3b  6a 2b 2  4ab3  b 4
(c) 81a 4  108a 3b  54a 2b 2  12ab3  b 4
(d) 81a 4  216a 3b  216a 2b 2  96ab3  16b 4
3.
(b) x 4  4 x3  6 x 2  4 x  1
(c) x 4  8 x 3  24 x 2  32 x  16
(a) x10  5 x8  10 x 6  10 x 4  5 x 2  1
(b) x15  10 x13  40 x11  80 x 9  80 x 7  32 x 5
(c) x8  8 x 2 
4.
108x 3
5.
160
6.
(a) 12 terms
7.
15120
8.
p  5
9.
 24
10.
c  3
24 32 16


x 4 x10 x16
(b) 28 160
© InThinking – IB Maths HL & SL
2