7a362

7xy, take 7cx

Ans. 8abc + b3a

14aby+7a2b2, take 9a2c14aby+15a2b2.

13 + 20ab3x — 46°cx2, take 31°cx2 + 9a6.x2

14. From 21x3y2 + 25x2y3 + 68xy1 40y5, take 64x2y3

+48ry — 40g.

Ans. 20.xy-39x2y3+21x3y2.

xy - 1363a

- 2cx

6xy.

3a2 -7c5d2

38. From what has preceded, we see that polynomials may be subjected to certain transformations.

These transformations consist in separating a polynomial into two parts, and then connecting the parts by the minus sign.

It will be observed that the sign of each term is changed when the term is placed within the parenthesis. Hence, if we have `one or more terms included within a parenthesis having the minus sign before it, the signs of all the terms must be changed when the parenthesis is omitted.

39. REMARK.-From what has been shown in addition and subtraction, we deduce the following principles.

1st. In Algebra, the words add and sum do not always, as in arithmetic, convey the idea of augmentation. For, if to a we add -b, the sum is expressed by ab, and this is, properly speaking, the arithmetical difference between the number of units expressed by a, and the number of units expressed by b. Consequently, this result is actually less than a.

To distinguish this sum from an arithmetical sum, it is called the algebraic sum.

Thus, the polynomial, 2a3-3a2b + 3b2c,

is an algebraic sum, so long as it is considered as the result of the union of the monomials

with their respective signs; but, in its proper acceptation, it is the arithmetical difference between the sum of the units contained in the additive terms, and the units contained in the subtractive term.

It follows from this, that an algebraic sum may, in the numer ical applications, be reduced to a negative expression.

2d. The words subtraction and difference, do not always convey the idea of diminution. For, the difference between

a and

is numerically greater than a. This result is an algebraic differ

ence,

40. It frequently occurs in Algebra, that the algebraic sign + or which is written, is not the true sign of the term before which it is placed. Thus, if it were required to subtract -b from a, we should write

Here the true sign of the second term of the binomial is plus, although its algebraic sign is -. This minus sign, operating upon the sign of b, which is also negative, produces a plus sign for b in the result. The sign which results, after combining the algebraic sign with the sign of the quantity, is called the essential sign of the term, and is often different from the algebraic sign.

MULTIPLICATION.

41. MULTIPLICATION, in Algebra, is the operation of finding the product of two algebraic quantities. The quantity to be multiplied is called the multiplicand; the quantity by which it is multiplied is called the multiplier; and both are called factors.

42. Let us first consider the case in which both factors are monomials.

Let it be required to multiply 7a3b2 by 4a2b; the operation may be indicated thus,

7a3b2 × 4a2b,

or by resolving both multiplicand and multiplier into their simple factors,

Taaabb4aab.

Now, it has been shown in arithmetic, that the value of a product is not changed by changing the order of its factors; hence, we may write the product as follows:

7 × 4aaaaabbb, which is equivalent to 28a5b3.

Comparing this result with the given factors, we see that the co-efficient in the product is equal to the product of the co-efficients of the multiplicand and multiplier; and that the exponent of each letter is equal to the sum of the exponents of that letter in both multiplicand and multiplier.

And since the same course of reasoning may be applied to any two monomials, we have, for the multiplication of monomials, the following

RULE.

1. Multiply the co-efficients together for a new co-efficient.

11. Write after this co-efficient all the letters which enter into the multiplicand and multiplier, giving to each an exponent equal to the sum of its exponents in both factors.

8. Multiply 5abd3 by 12cd5.

9. Multiply 7a4bd2c3 by abdc.

Ans. 60abcd8.

Ans. 7a5b2d3c4.

43. We will now proceed to the multiplication of polynomials. In order to explain the most general case, we will suppose the multiplicand and multiplier each to contain additive and subtractive terms.

Let a represent the sum of all the additive terms of the multiplicand, and b the sum of the subtractive terms; c the sum of the additive terms of the multiplier, and d the sum of the subtractive terms. The multiplicand will then be represented by ab and the multiplier, by cd.

-

We will now show how the multiplication expressed by (« — b) × (c — d) can be effected.

e times; for it is only the difference between a and b, that is first to be multiplied by c. Hence, ac bc is the product of

But the true product is a b taken c

d times: hence, the last product is too great by a b taken d times; that is, by ad bd, which must, therefore, be subtracted. Subtracting this from the first product (Art. 37), we have

(a - b) × (c — d) = ac bc ad + bd:

If we suppose a and c each equal to 0, the product will re duce to bd.

44. By considering the product of ab by cd, we may deduce the following rule for signs, in multiplication.

When two terms of the multiplicand and multiplier are affected with the same sign, their product will be affected with the sign +, and when they are affected with contrary signs, their product will be affected with the sign -.

or

We say, in algebraic language, that + multiplied by +, multiplied by —, gives +; multiplied by +, or + mul tiplied by -, gives-. But since mere signs cannot be multi plied together, this last enunciation does not, in itself, express a distinct idea, and should only be considered as an abbreviation of the preceding.

This is not the only case in which algebraists, for the sake of brevity, employ, expressions in a technical sense in order to secure the advantage of fixing the rules in the memory.

45. We have, then, for the multiplication of polynomials, the following

RULE.

Multiply all the terms of the multiplicand by each term of the multiplier in succession, affecting the product of any two terms with the sign plus, when their signs are alike, and with the sign minus, when their signs are unlike. Then reduce the polynomial result to its simplest form.