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If we consider the exponents of the preceding pow ers, we shall find that they follow a very simple law. Thus :

In the square, the exponents {

In the cube, the exponents

In the fourth power, the exponents

of a are 2, 1, 0,

of b are 0, 1, 2. S of a are 3, 2, 1, 0, of b are 0, 1, 2, 3. of a are 4, 3, 2, 1, 0, of b are 0, 1, 2, 3, 4,

etc., etc., etc.

In the first term of each power, a is raised to the required power of the binomial; and in the following terms the exponent of a diminishes by unity until we reach the last term which does not contain a.

The exponent of b in the second term is 1, and increases by unity in each term to the right, until we reach the last term in which the exponent is the same as that of the required power.

(131.) The sum of the exponents of a and b in any term is equal to the exponent of the given power. Thus, in the second power, the sum of the exponents of a and b in each term is 2; in the third power it is 3; in the fourth power it is 4, etc. This remark will enable us to detect any error so far as regards the exponents.

We hence infer that, for the seventh power, the terms without the coefficients must be

a', aob, a3b3, a1b3, a3b*, a2b3, abo, b';

QUEST.-What are the exponents of a and b in the different powers? What law do the exponents follow? What is the sum of the expe nents of a and b in any term?

and for the eighth power,"

ao, a'b, aob3, aob3,`a*b*, a3b3, a2bo, ab1, b'.

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The coefficient of the first term in every power of a+b is unity. The coefficient of the second term is the same as the exponent of the given power; and if the coefficient of any term be multiplied by the exponent of a in that term, and divided by the exponent of b increased by one, it will give the coefficient of the succeeding term.

Thus the fifth power of a+b is

a'+5a*b+10a3b'+10a2b3+5ab*+b°.

The coefficient of the second term is 5, which is the exponent of the power. Then, to find the coefficient of the third term, we multiply 5 by 4, the exponent of a in the second term, and divide by 2, which is the exponent of b increased by one. The quotient 10 is the coefficient of the third term. So, also, if 10, the coefficient of the third term, be multiplied by 3, the exponent of a, and divided by 3, the exponent of b increased by one, we obtain 10, the coefficient of the fourth term. Again if 10, the coefficient of the fourth term, be multiplied by 2, the exponent of a, and divided by 4, the exponent of b increased by one, we obtain 5, the coefficient of the fifth term.

The coefficients of the sixth power will also be found as follows:

6×5 15×4 20×3 15×2 6×1

1, 6,

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that is, 1, 6, 15,

20,

15,

6, 1.

;

QUEST.-What law do the coefficients follow?

The coefficients of the seventh power will be

7x6 21x5 35×4 35x3 21×2 7×1

1, 7,

2 3

4

5

6

7

that is,

1, 7, 21, 35,

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Therefore the seventh power of a+b is

a'+7a b+21a'b2+35a*b3+35a3b*+21a2b'+7ab®+b2. (133.) The following therefore, is the

BINOMIAL THEOREM.

In any power of a binomial a+b, the exponent of a begins in the first term with the exponent of the рош. er, and in the following terms continually decreases by one. The exponent of b commences with one in the second term of the power, and continually increases by one.

The coefficient of the first term is one; that of the second is the exponent of the power; and if the coefficient of any term be multiplied by the exponent of a in that term, and divided by the exponent of bincreased by one, it will give the coefficient of the succeeding term.

If we examine the powers of a+b in Art. 127, we shall find that after we pass the middle term, the same coefficients are repeated in the inverse order. Thus the coefficients of

(a+b)' are 1, 5, 10, 10, 5, 1;

of (a+b) are 1, 6, 15, 20, 15, 6, 1.

Hence it is only necessary to compute the coeffi cients for half the terms; we then repeat the same numbers in the inverse order.

QUEST.-Repeat the binomial theorem. Is it necessary to compute all the coefficients according to the rule?

Examples.

Ex. 1. Raise a+b to the 9th power.

The terms without the coefficients are

a', a3b, a'b3, aob3, a*b*, a*b3, a3bo, a3b′, ab3, b3. And the coefficients are

9x8 36x7 84x6 126x5 126x4 84x3

1, 9,

2' 3

4
36×2 9×1

5

6 7

8' 9

that is,

1, 9, 36, 84, 126, 126,

9, 1.

84,

36,

Prefixing the coefficients, we obtain (a+b)'=a'+9ab+36a*b*+84a°b3+126a*b*+126a*b*

+84a3b*+36a2b'+9ab'+bo.

It should be remembered that, according to a former remark, it is only necessary to compute the coefficients of half the terms independently.

Ex. 2. What is the seventh power of x+y?

Ans.

Ans

Ex. 3. What is the sixth power of x-a?

Ex. 4. What is the fifth power of m+n?

Ans.

(134.) If the terms of the given binomial are affected with coefficients or exponents, they must be raised to the required powers according to the principles already established for the involution of monomials.

Ex. 5. What is the fourth power of a-3b?

Ans.

QUEST. When the terms of the binomial have coefficients or expo ents, how do we proceed?

For convenience, let us substitute x for 3b; then

(a-x)'=a'-4a3x+6a2x2-4ax3+x*.

But a96'; x=276'; and x'=81b'. Substituting for x its value, we have (a-3b)'=a*-12ab+54a2b-108ab'+81b', Ans.

Ex. 6. What is the cube of 2a-3b?

Ans. 8a3-36a2b+54ab2-2763.

Ex. 7. What is the square of 5a-7b?

Ans. 25a-70ab+496'.

Ex. 8. What is the cube of 4x-5y?

Ans. 64x3-240x3y+300xy2- 125y3.

Ex. 9. What is the fifth power of x-3y?
Ans. x-15xy+90x3y2-270x3y3 +405xy*—243y'.
Ex. 10. What is the fourth power of 3a2-2b?

Ans. 81a-216a b+216a'b'-96a'b'+166'. Ex. 11. What is the fourth power of 2x+5a2? Ans. 16x+160x a2+600x2a'+1000xa +625a2. Ex. 12. What is the fourth power of 2x+4y?

Ans. 16x+128x3y+384x2y2+512xy3+256y' Ex. 13. What is the fourth power of a+b+c? Substitute x for b+c; then

(a+x)=a+4a3x+6a2x2+4ax3+x*.

Restoring the value of x, we have

(a+b+c)*=a*+4a3(b+c)+6a2(b+c)2+4a(b+c)3

+(b+c)*;

or, expanding the powers of b+c, we obtain (a+b+c)'=a*+4a3b+4a3c+6a2b2+12a2bc+6a2c2+4ab' +12ab'c+12abc2+4ac3+b*+4b3c+6b2c2+4bc3+c1. Ex. 14. What is the fourth power of a+b-2c? Ans. a*+4a3b-8a3c+6a2b2-24a2bc+24a2c2+4ab3

-24ab'c+48abc-32ac+ b'-8b'c+24b'c'-32bc'

+16c1.,

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