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38. In algebra the term “difference" does not always, as in arithmetic, denote a number less than the minuend: for, if from a we subtract – 6, the remainder will be a +b; and this is numerically greater than a. We distinguish between the two cases by calling this result the algebraic difference.

39. When a polynomial is to be subtracted from an algebraic expression, we inclose it in a parenthesis, place the minus sign before it, and then write it after the minuend. Thus, the expression 6 a’ — (3 ab 263 +2bc) indicates that the polynomial 3 ab — 263 +2bc is to be taken from 6a'. Performing the indicated operations by the rule for subtraction, we have the equivalent expression 6 a' – 3 ab + 262 2bc.

The last expression may be changed to the former by changing the signs of the last three terms, inclosing them in a parenthesis, and prefixing the sign – Thus,

6a’ – 3 ab + 262 — 2 bc=6a’ – (3 ab 262 +2bc). In like manner any polynomial may be transformed, as indicated below.

7a3 – 8 ab 46'c +663 = 7a - (8a2b + 46°c 683)

=7a3 – 8ab (46°c 663). 8a? 762 +0-d=8a? (762 – +d)

= 8a 73(-c+d). 963 – a + 3 a’ d=963 (a – 3a + d)

=963 - a-(-3a’+d). NOTE. — The sign of every term is changed when it is placed within a parenthesis which has the minus sign before it, and also when it is brought out of such parenthesis. 40. From the preceding principles we have

a-(+6)=a-6, and

a-(-6)=a+b. The sign immediately preceding b is called the sign of the quantity; the sign preceding the parenthesis is called the sign of operation; and the sign resulting from the combination of the signs is called the essential sign.

When the sign of operation is different from the sign of the quantity, the essential sign will be —; when the sign of operation is the same as the sign of the quantity, the essential sign will be +

MULTIPLICATION. 41. Multiplication is the operation of finding the product of two quantities.

The multiplicand is the quantity to be multiplied; the multiplier is that by which it is multiplied; and the product is the result. The multiplier and multiplicand are called factors of the product.

Exercises. 1. If a man earns a dollars in 1 day, how much will he earn in 6 days?

In 6 days he will earn six times as much as in 1 day. If he earns a dollars in 1 day, in 6 days he will earn 6 a dollars.

2. If i hat costs d dollars, what will 9 hats cost ?

Ans. 9 d dollars. 3. If 1 yard of cloth costs c dollars, what will 10 yards cost ?

Ans. 10c dollars. 4. If 1 cravat costs b cents, what will 40 cost ?

Ans. 406 cents. 5. If 1 pair of gloves costs b cents, what will a pairs cost?

If 1 pair of gloves costs b cents, a pairs will cost as many times b cents as there are units in a; that is, b taken a times, or ab, which denotes the product of b by a or of a by b.

6. If a man's income is 3 a dollars a week, how much will he receive in 46 weeks?

3a X 4b = 12 ab. If we suppose a = 4 dollars, and b = 3 weeks, the product will be 144 dollars.

NOTE. — It is proved in arithmetic (Davies' “Standard Arithmetic,” 8 50) that the product is not altered by changing the arrangement of the factors ; that is, 12 ab = u x b x 12 = b xa x 12 = a x 12 x b.

42. To find the Product of Two Positive Monomials.
Multiply 3 a62 by 2 ab.
We write,

3a2b2 x 2a2b = 3 x 2 x a? X a? x 62 x 6= 3 x 2 aaaabbb;
in which a is a factor 4 times, and b a factor 3 times : hence (8 14)

3a2b2 x 2 a2b = 3 x 2 a+b3 = 6 a*b, , in which we multiply the coefficients together, and add the exponents of the like letters.

The product of any two positive monomials may be found in like manner. Hence the rule:

Multiply the coefficients together, for a new coefficient.
Write after this coefficient all the letters in both monomials,

giving to each letter an exponent equal to the sum of its exponents in the two factors.

Exercises.

12xʻy

1. 8a2bc? x 7abd' = 56 aob?c?d?.
2. 21ab'cd x 8 abc= 168 a*b*c*d.

3. 4 abc x 7 df = 28 abcdf.
Multiply the following:-
4. 3a2b 6. 6.xyz 8. . 3 ab?co
2 ab

ayʻz

9 a?b%c 6abo 6 аху???

27 a85c 5. 12 aRx 7. aʼxy 9. 87 ax'y

2xy? 36°x*y: 144 a?:coy 2aco 261 abpxc@y" 10. 5ab?x? by 6 c*2.6.

Ans. 30a8b%c92%. 11. 10 a*b5c8 by 7 acd.

Ans. 70 2555cd. 12. 36 a®6"c®dby 20 ab?cod*.

Ans. 720 aoboc°do. 13. 5am by 3ab".

Ans. 15 am+16". 14. 3amk3 by 6 a2b".

Ans. 18 am+28m+5. 15. 6ambn by 9a%b?.

Ans. 54 am+5%+7 16. 5amin by 2 apb!

Ans. 10 am+pbn+e. 17. 5aml?c? by 2 ab"c.

Ans. 10 am+ifn+20. 18. 6a?bmen by 3 a?l?c?.

Ans. 18a%bm+?cm+? 19. 20 aʼb%cd by 12 aʼxy.

Ans. 240 a'b%cdxảy. 20. 14 a*b*d*y by 20 a'coxy.

Ans. 280 a?b%c%d*2*y?. 21. 8a+bya by 7a'bxyo.

Ans. 56 a?b*xy".

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From the preceding examples we deduce the following rule:

When the factors have like signs, the sign of their product will be t.

When the factors have unlike signs, the sign of their product will be

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