find the coefficient of x¹¹ in expression of (x⁴-1/x³)¹⁵
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Answer:
First expand the term (1+2x)
4
by binomial expansion.
(1+2x)
4
=
4
C
0
(1)
4
(2x)
0
+
4
C
1
(1)
3
(2x)
1
+
4
C
2
(1)
2
(2x)
2
+
4
C
3
(1)
1
(2x)
3
+
4
C
4
(1)
0
(2x)
4
=1+8x+24x
2
+32x
3
+16x
4
(1)
Now expand the term (2−x)
5
by binomial expansion,
(2−x)
5
=
5
C
0
(2)
5
(x)
0
−
5
C
1
(2)
4
(x)
1
+
5
C
2
(2)
3
(x)
2
−
5
C
3
(2)
2
(x)
3
+
5
C
4
(2)
1
(x)
4
−
5
C
5
(2)
0
(x)
5
=32−80x+80x
2
−40x
3
+10x
4
−x
5
(2)
Multiply the coefficients of those powers which can give the term x
4
and then add from equation (1) and (2).
=1×10+8(−40)+24(80)+32(−80)+16(32)
=−438
Therefore, the coefficient of x
4
is −438.
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