CHAPTER IL.

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION.

ADDITION.

31. ADDITION, in algebra, is the operation of finding the simn. plest equivalent expression for the aggregate of two or more algebraic quantities. Such equivalent expression is called their sum.

32. If the quantities to be added are dissimilar, no reductions can be made among the terms. We then write them one after the other, each with its proper sign, and the resulting polynomial will be the simplest expression for the sum. For example, let it be required to add together the monomials

3a, 56 and 2c;

we connect them by the sign of addition,

3a +56 +2c,

a result which cannot be reduced to a simpler form.

33. If some of the quantities to be added have similar terms, we connect the quantities by the sign of addition as before, and then reduce the resulting polynomial to its simplest form, by the rule already given. This reduction will, in general, be more readily accomplished if we write down the quantities to be added, so that similar terms shall fall in the same column. Thus ;

Let it be required to find the sum of the quantities,

Their sum, after reducing (Art. 29), is

3a2

4ab

2a2 3ab + b2

{

5a2

2ab 562

5ab 462

34. As operations similar to the above apply to all algebraic expressions, we deduce, for the addition of algebraic quantities, the following general

RULE.

1. Write down the quantities to be added, with their respective. signs, so that the similar terms shall fall in the same column.

II. Reduce the similar terms, and annex to the results those terms which cannot be reduced, giving to each term its respective sign.

EXAMPLES.

1. Add together the polynomials,

3a2 - 2b2-4ab, 5a2 — b2+2ab and 3ab 3c2

The term 3a2 being similar to 5a2 we write 8a2 for the result of the reduction of these two terms, at the same time slightly crossing them as in the terms of the example.

3&2

-

262.

4&b 282

5x2+2ub

+3ab

8a2+ ab 562 3c2

-

Passing then to the term -4ab, which is similar to the two terms +2ab and +3ab, the three reduce to ab, which is placed after 8a2, and the terms crossed like the first term. Passing then to the terms involving b2, we find their sum to be -562, after which we write - 3c2.

The marks are drawn across the terms, that none of them may be overlooked and omitted.

6. Add together 3a + b, 3a +36, 9a7b, 6a+ 96 and 8a+ 3b+8c. Ans. 11a +96 +8c.

7. Add together 3ax + 3ac+f, -9ax+7a+d, + 6ax +3ac +3f, 8ax +13ac + 9f and 14ƒ+ 3ax.

--

Ans. 11ax+19xc −ƒ + 7a + d.

8. Add together the polynomials, 3a2c + 5ab, 7a2c — 3ab + 3ac 5a2c bab+9ac and - Sa2c + ab- - 12ac. Ans. 7a2c 3ab. 9. Add the polynomials, 19a2x3b12a3cb, 5a2x3b+15a3cb 10ax, -2a2x3b-13a3cb and 18a2x3b12a3cb + 9 ax.

10. Add together 3a+b+c, 5a +2b+3ac, a + c + ac and

- За -9ac-8b.

Ans. 6a 5b2c5ac.

11. Add together 5a2b+6cx + 9bc2, 7cx

- 9bc2 + 2a2b.

· 18ax +6a2 + 10ab

Ans. ·3a2b2c2 + 6a2.

13. What is the sum of 41a3b2c - 27abc-14a2y and 10a3b2c +9abc? Ans. 51a3b2c 18abc-14a2y.

and 10ab16a2b2c2+ 19a3x. Ans. 10a2+13a2b2c2 + 10ab.

20. Add together 7a2b 3abc 862c

362c14c3+2cd2 3d3.

21. Add together — 18a3b+2ab+6a2b2, — Saba +7a3b — 5a2b2 and - 5a3b + 6ab1 + 11a2b2. Ans. - 16a3b+12a2b2.

22. What is the sum of 3a3b2c16a4x-9ax3d, + 6a3b2c - Sax3d + 17a1x and +16ax3d — a*x

8a3b2c?

Ans. a3b2c+ ax3d.

23. What is the sum of the following terms: viz., 8a5 10a4b

163624a2b3 12ab15a3b2 + 24a2b3 20263 +32ab4 865?

6ab4

Ans. 8a522a4b17a3b2 + 48a2b3 + 26ab1.

16a3b2

865.

SUBTRACTION.

35. SUBTRACTION, in algebra, is the operation for finding the simplest expression for the difference between two algebraic quantities. This difference is called the remainder.

36. Let it be required to subtract 46 from 5a. Here, as the quantities are not similar, their difference can only be indicated, and we write

Again, let it be required to subtract 4a3b from 7a3b. These terms being similar, one of them may be taken from the other, and their true difference is expressed by

37. Generally, if from one polynomial we wish to subtract another, the operation may be indicated by enclosing the second in a parenthesis, prefixing the minus sign, and then writing it after the first. To deduce a rule for performing the operation thus indicated, let us represent the sum of all the terms in the first polynomial by a. Let c represent the sum of all the ad ditive terms in the other polynomial, and -d the sum of the subtractive terms; then this polynomial will be represented by cd. The operation may then be indicated thus,

where it is required to subtract from a the difference between c and d

If, now, we diminish the quantity a by the quantity c, the result ac will be too small by the quantity d, since c should have been diminished by d before taking it from a. Hence, to obtain the true remainder, we must increase the first result by d, which gives the expression

a-c+d,

and this is the true remainder.

By comparing this remainder with the given polynomials, we see that we have changed the signs of all the terms of the quantity to be subtracted, and added the result to the other quantity. To facilitate the operation, similar quantities are written in the same column.

Hence, for the subtraction of algebraic quantities, we have the following

RULE.

I. Write the quantity to be subtracted under that from which it is to be taken, placing the similar terms, if there are any, in the same column.

II. Change the signs of all the terms of the quantity to be subtracted, or conceive them to be changed, and then add the result to the other quantity.