1. (a+b)3 . . a3+a2b+ab2+b3.

2. (a-b)1.. a1-a3b+a2b2-ab3+b4.

3. (a+b)3 ̧ . . a5+aab+a3b2+a2b3+aba+b5.

4. (a—b)1‚‚ a2-a©b+a3b2—a1b3+a3b1—a2b3+ab¤—b7.

Of the Coefficients.

The co

89. The coefficient of the first term is unity. efficient of the second term is the same as the exponent of the given power. The coefficient of the third term is found by multiplying the coefficient of the second term by the exponent of the leading letter, and dividing the product by 2. And finally—If the coefficient of any term be multiplied by the exponent of the leading letter, and the product divided by the number which marks the place of that term from the left, the quotient will be the coefficient of the next term. Thus, to find the coefficients in the example

(a—b)... a1—a©b+a3b2—a1b3+a3b1 — a2b3+ab®- b7

we first place the exponent 7 as a coefficient of the second term. Then, to find the coefficient of the third term, we multiply 7 by 6, the exponent of a, and divide by 2. The quotient 21 is the coefficient of the third term. To find the coefficient of the fourth, we multiply 21 by 5, and divide the product by 3: this gives 35. To find the coefficient of the fifth term, we multiply 35 by 4, and divide the product by 4 this gives 35. The coefficient of the sixth term, found in the same way, is 21; that of the seventh, 7; and that of the eighth, 1. Collecting these coefficients, we

have

:

(a-b)"=

a2 —7aob—21a5b2 —35aab3+35a3ba—21a2b5+7ab6—b2.

REMARK. We see, in examining this last result, that the coefficients of the extreme terms are each unity, and that he coefficients of terms equally distant from the extreme terms are equal. It will, therefore, be sufficient to find the coefficients of the first half of the terms, from which the others may be immediately written,

EXAMPLES.

1. Find the fourth power of a+b.

Ans. a4a3b+6a2b2+4ab3+64.

2. Find the fourth power of a-b.

Ans. a-4a3b+6a2b2-4ab3+64.

3. Find the fifth power of a+b.

Ans. a5+5a+b+10a3b2+10a2b3+5ab4+b5.

4. Find the fifth power of a-b.

Ans. a5-5a4b+10a3b2—10a2b3+5ab1—b5,

5. Find the sixth power of a+b.

Ans. a6+6a5b+15a+b2+20a3b3+15a2b1‡6ab5+bo.

6. Find the sixth power of a-b.

Ans. a6—6a5b+15a4b2—20a3b3+15a2ba—6ab5+bo. 7. Let it be required to raise the binomial 3a2c-2bd to the fourth power.

It frequently occurs that the terms of the binomial are affected with coefficients and exponents, as in the above

QUEST.-89. What is the coefficient of the first term? What is the coefficient of the second? How do you find the coefficient of the third term? How do you find the coefficient of any term? What are the coefficients of the first and last terms? How are the coefficients of terms equally distant from the extremes?

example. In the first place, we represent each term of the binomial by a single letter. Thus, we place

and

x3-27a6c3,

x4=81a8c4;

y2=4b2d2, y3 — — 8b3d3, y416b4d1.

Substituting for x and y their values, we have (3a3c—2bd)1 = (3a2c)4+4(3a2c)3(−2bd)+6 (3a2c)2 (—2bd)2 +4(3a2c) (-2bd)3+(−2bd)1‚·

and by performing the operations indicated, (3a3c-2bd)=81a8c4-216a6c3bd +216a4c2b2d2-96a2cb3d3

11. What is the eighth power of m+n?

Ans. m8+8min +28m6n+56m3n3 + 70m*n*+56m3n5.

+28m2n6+8mn"-+n8.

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

Ans. a1-12a3b+54a2b2-108ab3+81b1.

13. What is the fifth power of c-2d?

Ans. c5-10c4d+40c3d2-80c2d3 +80cd4—32d5.

14. What is the cube of

5a-3d?

Ans: 125a3-225a2d+135ad2 —27d3.

REMARK. The powers of any polynomial may easily be found by the Binomial Theorem.

15. For example, raise a+b+c to the third power.

Then, (a+b+c)3=(a+d)3=a3+3a2d+3ad2+d3.

Or, by substituting for the value of d,

This expression is composed of the cubes of the three terms, plus three times the square of each term by the first powers of the two others, plus six times the product of all It is easily proved that this law is true for any

three terms. polynomial.

To apply the preceding formula to the development of the cube of a trinomial, in which the terms are affected with coefficients and exponents, designate each term by a single letter, then replace the letters introduced, by their values, and perform the operations indicated.

From this rule, we find that

(2a2-4ab+3b2)3=8a6-48a5b+132a4b2-208a3b3

+198-108ab5+2766.

The fourth, fifth, &c, powers of any polynomial can be found in a similar manner.

16. What is the cube of a-2b+c?

Ans. a3-8b3+c3—6a2b+3a2c+12ab2+1262c+3ac2

-6bc2-12abc.

CHAPTER V.

Extraction of the Square Root of Numbers. Formation of the Square and Extraction of the Square Root of Algebraic Quantities. Calculus of Radicals of the Second Degree.

90. The square or second power of a number, is the product which arises from multiplying that number by itself once for example, 49 is the square of 7, and 144 is the square of 12.

91. The square root of a number is that number which, being multiplied by itself once, will produce the given number. Thus, 7 is the square root of 49, and 12 the square root of 144 for, 7x7=49, and 12×12=144.

92. The square of a number, either entire or fractional, is easily found, being always obtained by multiplying this number by itself once. The extraction of the square root of a number is, however, attended with some difficulty, and requires particular explanation.

QUEST.-90. What is the square, or second power of a number?— 91. What is the square root of a number?