SECTION VI.

MULTIPLICATION.

46. MULTIPLICATION is a short method of finding the sum of the repetitions of a quantity.

47. The multiplier must always be an abstract number, and the product is always of the same nature as the multiplicand.

The cost of 4 pounds of sugar at 17 cents a pound is 17 cents taken, not 4 pounds times, but 4 times; and the product is of the same denomination as the multiplicand 17, viz. cents.

In Algebra the sign of the multiplier shows whether the repetitions are to be added or subtracted.

1.

(+ a) × (+4)=+4a;

i. e. a added 4 times is +a+a+a+a+4 a.

2.

i. e. + a subtracted 4 times is

3.

i. e.

4.

i. e.

a subtracted 4 times is +a+a+a+a= + 4 a.

In the first and second examples the nature of the product is ; in the first, the sign of 4 shows that the product is to be added, and + 4 a added is + 4 a; in the second, the sign of 4 shows that the product is to be subtracted, and + 4 a subtracted is the third and fourth examples the nature of the product is; in the third, the sign of 4 shows that the product is to be added, and

4 a.

In

4 a added is

4a; in the

subtracted, and

sign of 4 shows that the product is to be

- 4 a subtracted is + 4 a.

48. Hence in multiplication we have for the sign of the product the following

RULE.

Like signs give +; unlike, -.

Hence the products of an even number of negative factors is positive, of an odd number, negative.

49. Multiplication in Algebra can be presented best under three cases.

CASE I.

50. When both factors are monomials.

1. Multiply 3 a by 2b.

OPERATION.

3 a X 2b= 3 × a × 2 × b = 3 × 2 × a xb6ab.

As the product is the same in whatever order the factors are arranged, we have simply changed their order and united in one product the numerical coefficients.

Hence, when both factors are monomials,

RULE.

Annex the product of the literal factors to the product of their coefficients, remembering that like signs give + and unlike,

2. Multiply a3 by a2.

OPERATION.

a3× a2 = (a XaX a) × (a × a) = a×a×a×a×a = a3

α

As the exponent of a quantity shows how many times it is taken as a factor, a3 = axaxa; and a2 = axa; and a3 X a2 = ax a xa xa xa, and this is equal to a3. (Art. 24.) Hence,

Powers of the same quantity are multiplied together by adding their exponents.

17. Multiply together 444 x y, 3 x2 y3, and 2 z.

2664x3 y1 z.

Ans. 30m+n

5 a2bc, and 4 a b2.

22. Multiply 4 (x + y) by 3 (x + y).

Ans. 75 bxay.

Ans. 12 (x + y)2.

NOTE. Any number of terms enclosed in a parenthesis may be treated as a monomial.

25. Multiply 4 (a + b)m by 2 (a + b)".

Ans. 8 (a+b)m+n.

26. Multiply a3 (x + z)2 by a b2 (x + z).

CASE II.

51. When only one factor is a monomial.

1. Multiply 8 +5 by 3.

times, or 15, too much; the product required, therefore, is 24

Therefore,

15.

The product of the sum is equal to the sum of the prod ucts, and the product of the difference to the difference of the products.

3. Multiply x+y-z by a.

Multiply each term of the multiplicand by the multiplier, and connect the several results by their proper signs.

6. Multiply 5 mn + 4 m2-6n2 by 4 a b.

7. Multiply 16 a2 x 8 x z + 4y by

8. Multiply bx3-cx2 + dx by x3.

-

3x y.

9. Multiply 63 x y

Ans. 252 x y z + 56 x z + 24 z2. 10. Multiply 14 a1 — 13 a3 + 12 a2 - 11 a by 4 a3.

11. Multiply x-2a+14 by a x.

12. Multiply 17 ax 14 by + 11 cz by 4 a bé xyz.

13. Multiply 21 a2b2-3x y2-4bc by-9axy.

Multiply each term of the multiplicand by each term of the multiplier, and find the sum of the several products.