cal difference between the sum of the units contained in the additive terms, and the sum of the units contained in the subtractive terms.

It follows from this, that an algebraic sum may, in the numerical applications, be reduced to a negative number, or a number affected with the sign.

2d. The words subtraction and difference, do not always convey the idea of diminution. For, the difference between + a and - b being a (b) = a + b, is numerically greater than a.

:

result is an algebraic difference.

This

MULTIPLICATION.

40. ALGEBRAIC multiplication has the same object as arithmetical, viz., to repeat the multiplicand as many times as there are units in the multiplier. The multiplicand and multiplier are called factors.

It is proved in Arithmetic (see Davies' Arithmetic, § 22), that the value of a product is not affected by changing the order of its factors that is,

12ab ab x 12 = ba x 12 = a × 12 × b.

For convenience, however, the letters in each term are generally arranged in alphabetical order, from the left to the right.

Let it be required to multiply 7a3b2 by 4a2b.

By decomposing the multiplicand and multiplier into their factors, we may write the product under the form

7a8b2 × 4a2b = 7aaabb × 4aub;

and since we may change the order of the factors without affecting the value of the product, we have,

7a3b2 × 4a2b= 7 × 4aaaaabbb == 28a5b3;

a result which is obtained by multiplying the co-efficients together for a new co-efficient, and adding the exponents of the same letter, for the new exponents.

Again: multiply the monomial 12a2b4c2 by 8a3b2d2.

We can place the product under the form,

12a2b4c2 x 8a3b2d2 =

12 x 8aaaaabbbbbbccdd = 96a5b6c2d2.

By considering the manner in which these results are obtained, we see, that any quantity, as a, must be found as many times

a factor in the product, as it is a factor in both the multiplicand and multiplier; which number will always be expressed by the sum of its exponents.

41. Hence, for the multiplication of monomials we have the following

RULE.

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

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

42. 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 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 ub and the multiplier, by c d.

We will now show how the multiplication expressed by d) can be effected.

(a - b) x (c

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

b taken c

by c. But the true product is a
the last product is too great by a
by ad
- bd, which must be subtracted.

- d times: hence,

- b taken d times; that is,

Changing the signs and

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 reduce to + bd.

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

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 Again, we say in

or multiplied by

-.

algebraic language, that + multiplied by +, gives +; multiplied by +, or + multiplied by, gives gives - But since mere signs cannot be multiplied 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.

44. Hence, for the multiplication of polynomials we have 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.

1. Multiply by

The product after reducing,

becomes

3a2 + 4ab + b2

2a + 5b

6a3 + 8a2b+2ab2

+15a2b+20ab2 + 5b3

6a3 +23a2b + 22ab2 + 5b3.

6. Multiply 3ab2 + 6a2c2 by 3ab2+3a2c2.

7. Multiply 4x2 - 2y by 2y.

Ans. 9a2b4 + 27a3b2c2 + 18a4c4.

8. Multiply 2x+4y by 2x - 4y.

Ans. 8x2y - 4y2.

Ans. 4x2 16y2.

y. Ans.

9. Multiply x3 + x2y + xy2 + y3 by x
10. Multiply a2 + xy + y2 by x2xy + y2.

Ans. x + x2y2 + y1.

In order to bring together the similar terms, in the product of two polynomials, we arrange the terms of each polynomial with reference to a particular letter.

After having arranged the polynomials, with reference to the letter a, multiply each term of the first, by the term 2a2 of the second; this gives the polynomial 8a5-10ab16a3b2 + 4a2b3, the signs of which are the same as those of the multiplicand. Passing then to the term 3ab of the multiplier, multiply each term of the multiplicand by it, and as it is affected with the sign affect each product with a sign contrary to that of the corresponding term in the multiplicand; this gives

The same operation is also performed with the term - 462, which is also subtractive; this gives,

The product is then reduced, and we finally obtain, for the most simple expression of the product,

8a5 22a+b - 17a3b2+48a2b3 + 26ab1 .

12. Multiply 2a2

865.

3ax+4x2 by 5a2 . 6ах - 2x2. Ans. 10a 27a3x+34a2x2 18ax3 8x4.

13. Multiply 3x2-2yx+5 by x2+2xy-3.

Ans. 3x+4x3y - 4x2 - 4x2y2 + 16xy - 15. 4x2y2 ·

14. Multiply 3x3 + 2x2y2+3y2 by 2x3-3x2y2+5y3.

6x65x5у26x4y+6x3y2+15x33 9x2y++10x2y5 + 15y5.

Ans.

{

Prod. red.

-5a24bd8cf

15a37a2bd · 29a2cf — 20b2d2 + 44bcdf — 8c2f2.

45. Results deduced from the multiplication of polynomials. 1st. If the polynomials which are multiplied together are homogeneous,

Their product will also be homogeneous, and the degree of each term will be equal to the sum of the degrees of any two terms of the multiplicand and multiplier.

Thus, in example 18th, each term of the multiplicand is of the 5th degree, and each term of the multiplier of the 3d degree: hence, each term of the product is of the 8th degree. This remark serves to discover any errors in the addition of the exponents.