082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

6.4 The binomial theorem 217
EXERCISES6.4
1 Usethe binomial theorem to expand
(a) (1 +x) 3 (b) (1 +x) 4 (c)
(d)
(
1 − x 2) 5
(e)
(
2 + x 2) 5
(f)
2 UsePascal’striangleto expand (a +b) 8 .
(
(
1 + x 3
3 − x 4
3 UsePascal’striangleto expand (2x +3y) 4 .
4 Expand (a −2b) 5 .
5 Usethe binomial theoremto findthe expansion of
(3 −2x) 6 upto andincluding the term inx 3 .
6 Obtainthe firstfour terms in the expansion of
(
1 + 1 2 x ) 10
.
7 Obtainthe firstfive termsin the expansion of
(1 +2x) 1/2 .Statethe rangeofvalues ofxforwhich
the expansion is valid.Chooseavalue ofxwithin the
rangeofvalidity andcomputevalues ofyour
expansionforcomparison with the truefunction
values. (
8 Expand 1 + 1 ) −4
2 x in ascending powersofxupto
the term inx 4 ,stating the range ofvalues ofxfor
whichthe expansion is valid.
(
9 Expand 1 + 1 −1/2
in descending powers upto the
x)
fourthterm.
) 4
) 4
10 (a) Expand (1 +x 2 ) 4 .
(b) Expand (1 +1/x 2 ) 4 .
11 A function, f (x),isgiven by
( ) 1/2
f(x)=
1 + 1 x
(a) Obtain the firstfour termsin the expansion of
f (x)in descending powersofx.State the range
ofvalues ofxforwhich the expansion is valid.
(b) By writing f (x)in the form
f(x) =x −1/2 (1 +x) 1/2
obtain the firstfourterms in the expansion of
f (x)in ascending powersofx.State the range of
values ofxforwhich the expansion is valid.
12 The function,g(x),is defined by
g(x) =
1
(1+x 2 ) 4
(a) Obtain the firstfour termsin the expansion of
g(x) in ascending powersofx. Statethe range of
values ofxforwhich the expansion is valid.
(b) By rewritingg(x) in an appropriate form,obtain
the firstfour termsin the expansion ofg(x) in
descending powers ofx.State the range ofvalues
ofxforwhich the expansion isvalid.
Solutions
1 (a) 1+3x+3x 2 +x 3
3 16x 4 +96x 3 y +216x 2 y 2 +216xy 3 +81y 4
(b) 1+4x+6x 2 +4x 3 +x 4
4 a 5 −10a 4 b +40a 3 b 2 −80a 2 b 3 +80ab 4 −32b 5
(c) 1+ 4x
3 + 2x2
3 + 4x3
27 + x4
5 729 −2916x +4860x 2 −4320x 3
81
(d) 1− 5x
2 + 5x2
2 − 5x3
4 + 5x4
16 − x5 6 1+5x+ 45x2 +15x 3
32
4
(e) 32 +40x +20x 2 +5x 3 + 5x4
8 + x5 7 1+x− x2
32
2 + x3
2 − 5x4
8 valid for −1 2 <x< 1 2
(f) 81 −27x + 27x2 − 3x3
8 16 + x4 8 1−2x+ 5x2
256
2 − 5x3
2 + 35x4 valid for −2 <x<2
16
2 a 8 +8a 7 b +28a 6 b 2 +56a 5 b 3 +70a 4 b 4 +56a 3 b 5 + 9 1 − 1
28a 2 b 6 +8ab 7 +b 8 2x + 3
8x2 − 5
16x 3