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a', -a'r, ta'x', -a'r', ta't*, -ar", tx'. Therefore the expansion will be

(a-x'=a-6°x+15a'x_20a'x' +15a'x!-har+x. 2. Expand (a +x)# into a series.

In this example n=1 Represent the coefficients by A, B, C, D.... ; then A=

1 B=AX

+1

=+2

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5

a

a

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The literal factors of the terms will be a tx

,-*,.... Hence, (a+x)} 1 3

3.5
x +

a
2.4
2:4.6

2:4.6.8
or by taking out the factor at, in the second member,
(a+2) =

1
3

3.5 a? (1+ £a***

a x2 +

ar or*zo+....) 2:1:6

2-4.6.8 or by clearing of negative esponents,

x* 3.28

3.5.x* (a+x)} = a(1+za

+

+...

....) 2:19 2:4.64 2:4.6.82* We might have obtained this last result directly, by putting the binomial in the form of a (1+

a:(1+) It is well, however, to note the transformations made above.

2.4

into a series. (a+x)"

Observe that

3. Expand cat
(0+2) = (2+2)* = a*(1+3)* = *(1+3)*

(1+)*

-....)

Whence, by expanding the factor (1+

we obtain

22c

+

a

1 1

3.x* 4.8 5.2

1 + (a+x)

a as 4. Expand (a'-**) into a series.

If we take the descending powers of a', commencing with the 5th, and the ascending powers of *, commencing with the first, we have for the literal factors of the terms,

a', a"x", a'x', a®x, a®z, ". Hence, with the coefficients the development becomes

(a'—) = a_5a""+10a'x*—10aoxo+5a'x'_".

EXAMPLES FOR PRACTICE.

1. Find the fifth power of a_b.

Ans. a'-5a*6+10a'/'-10a'b' +5ab-6. 2. Find the sixth power of 1+c.

Ans. 1+6c+15c+20c +15c* +6c+c'. 3. Find the seventh power of x+y. Ans. x'+7coy+21xoyo +35x*y* +35x*y*+21x*yo +7xyo+y'. 4. Find the eighth power of a'–1.

Ans. a16_8a" +28a" —56a"' +70a-564+28a-8a*+1. 5. Find the ninth power of a-c.

Ans. a'—9a%c + 36a'c'—84a® + 126ac4_126a*c* + 84ac_ 36a'c'+9ac-c. 6. Expand (1+ax)'.

Ans. 1+5ax+10aʻx® +10aox' +5a*x* +aʻxco. 7. Expand (a'—*).

Ans a' _6a^°** +15aRx_20aox® +15a*«c°—6aRx+0 +*".

8. Expand (x*—2).

Ans. "_5.xoz*+10x®z" - 10.0*21 +5x®z_2*. 9. Expand (a's+dy').

Ans. a'c' + 6a"ex'dy + 15a'x'doy+20aox'doy' +15a x'd'y + 6a'xd"y"+doy". 10. Expand (a—x)# into a series.

wa
3x

3 · 5x Ans.

2a 2 · 4a 2·4·6a 2.4.6.8a

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15. Expand 1–0)

.

into a series.

Ans. a(1+2x+3x+ 4x® + 5x + 6.09 +....). 16. Expand (a+b)into a series. 79 34

386

3.50
Ans. at
+

to... 2n 2 · 4a 2.4.6ao 2.4.6.8a'

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19. Expand (1-a)-3 into a series.

Ans. 1+30+ 6a’ + 10a' +152+212 +28ao +36a' +.... 20. Expand (a —X*) into a series.

3.x 3x* 3.5x 3.5.9x Ans. Vaa

4a 4.8a* 4.8.12a 4.8.12.16a' 21. Expand (a+y)-4 into a series.

1 4y 10y 20y 35y* 56y
Ans.
a
+
+

to... as a

al

ya (a

....) 22. Expand VT

into a series.

22 670 6•117* 6.11.1676 Ans. r +

+
5

+ +
2-5 2:3-5 2-3-4.50

+.... 23. Expand "V1_&* into a series.

14.x® 14.29x": 14.29-442" Ans. 1

15 2.159 2.3.15 2.3.4.15

METHOD OF SUBSTITUTION.

378. In the formula (c+y)*= n(n − 1)

n(n 1)(n-2) 20*+nx-y+ c"-" +

ayt.... 2

2 : 3 we may suppose x and y to represent any quantities whatever; and thus we may obtain the development of the powers of binomials with numerical coefficients, or of polynomials.

1. Involve 3a+2c to the fifth power.
The binomial coefficients for the fifth power are

1, 5, 10, 10, 5, 1. And by connecting these with the powers of the given terms, according to the law of the formula, we have

(3a+2c) = (3a)+5(3a) (2c)+10(3a)'( 2c)' + 10(3a)'(20)' + 5(3a)(20)+(2c)'; or, by performing the operations indicated,

(3a+20) = 243ao+810a*c+1080a'c'+720a'c'+240aco +32c

2. Involve a +6+2c' to the fourth power.

We may consider the polynomial in two parts, atb, represented by x, and +2c represented by y. Then we have

(a + b + 2c*)* = (a +1)*+4(a+0)'(2c) + 6(a+b)'(2c)' + 4(a+b) (2c)+(2c).

Performing the operations indicated,

(a+b+2c*)*=a*+ 4aRb+ 6a2bo+4ab+6*+8a'c*+24a'bc+ 24aboco+86*c+24a'c*+48abc* +246"c* +32aco +326co +16c'.

EXAMPLES FOR PRACTICE.

1. Find the third power of a--2b.

Ans. a'-_-6a'b+12ab-86'. 2. Find the fourth power of 2a+3x.

Ans. 16a +96a*x+216a'.x' +216ax+81x*. 3. Find the fourth power of 1-a.

Ans. 1—2a+kama+za'. 4. Find the fourth power of a'-ax+co.

Ans, a® — 4a’x + 10uoix— 16u*x + 19a*x* — 16a®X® +- 10aox' — 4ax' + 20%. 5. Expand (4a*—3x)# into a series.

27c" 567x Ans. V 2a 1

16a” 512a" 24576a"

32

Vzali

.....)

FRENCH'S THEOREM.

379. When a binomial having numerical coefficients is to be raised to any power, the coefficients of the expansion may be obtained with great facility by means of a simple modification of the binomial formula. We have (z+u)" =

n(n-1) n(n-1)(n-2) Z"+nz-u+

-734+.... 2

2 3 In this equation make >= ax, and u=

by; then (ax+by)=a"x"+nan-16.00hyt

-1) n(n

a*-**.**y+

2 n(n-1)(1-2)

an-888.cmdy' to... 2 3

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