SUBTRACTION.

56. SUBTRACTION, in Algebra, is the process of finding the difference between two algebraic quantities.

The Subtrahend is the quantity subtracted.

The Minuend is the quantity from which it is subtracted. The Difference, or Remainder, is the quantity left after the subtraction is performed.

In algebraic subtraction there may be distinguished two cases.

CASE I.

57. When the quantities are positive.

1. Let it be required to take 3 a from 8 a.

OPERATION.

8 a

3 a 5 a

Since 8 times any quantity less 3 times the quantity equals 5 times the quantity, 8a less 3 a equals 5 a.

2. Let it be required to take 86 from 56.

OPERATION.

5 b

8b=

We cannot, numerically take 3b 8 times any quantity from 5 times the same quantity. If

we take 56 from 5b, nothing will remain; there is yet, however, a quantity, 3b, to be subtracted, with nothing to take it from, which, from its nature (Art. 34), we indicate by

- 3 b.

3. Let it be required to take b+c from a.

OPERATION.

a (b+c)=abc

If b is taken from a, we have a - b. But a is to be diminished by

c, as well as b, consequently the true remainder will be

CASE II.

58. When the quantity to be subtracted is partly,

away c too much, consequently the true difference will be α- b increased by c, or a

Hence,

·b+c.

The ALGEBRAIC DIFFERENCE between two quantities may be numerically greater than either quantity.

The term difference, in algebra, it will be observed, has a much more general signification than in arithmetic; the same is also equally true of the term sum (Art. 53).

59. It follows, from the two preceding cases, that

Subtracting a positive quantity is the same as adding an equal negative quantity, and subtracting a negative quantity is the same as adding an equal positive quantity.

Hence, the following

GENERAL RULE.

Conceive the signs of all the terms of the subtrahend to be changed, from + to, or from to +, and then proceed as in addition.

17x2

9. From 312-3y2+ab take 17 x2+5y2-4ab+7. 10. Take a b from a + b.

Ans. 2b.

11. Take 6a3b5c from 6a+36-5c+1.

12. From 5xy+4a2 662 take 3xy-9 a2 — b2 + 4. 13. From 11a7b+c take a +76-3c+11.

Ans. 10 a 14b4c11.

60.

It is sometimes sufficient to indicate the subtraction of a polynomial, by inclosing it in a parenthesis, and prefixing the sign. Thus,

5 a2 (a2 + b2 c)

indicates that the entire quantity a2 + b2. c is to be taken

from 5 a2.

Performing the operation indicated, by the rule, Art. 59, we have

5 a2 a2

b2 + c, or 4 a2

b2 + c.

The first of these expressions may be transformed to its previous equivalent form, by changing the signs of the last three terms and inclosing them in a parenthesis, with the sign prefixed. Hence,

When an expression within a parenthesis is preceded by the sign, the parenthesis may be removed, if the sign of every inclosed term be changed; and

Any number of terms in an expression may be inclosed by parenthesis and the sign prefixed, if the sign of every term

inclosed be changed.

18.

Indicate the subtraction of 6 a + b2 from 7 a - b.

22. Remove the parenthesis from the expression 6 a -9x (4a-5x).

23. Remove the parentheses from 4a 5x (ax) +(x — 8a).

Ans. - 5 a.

24. Place in a parenthesis, with the negative sign prefixed, the last three terms of the expression 2a3-3a2b +4ab2 - a3 — b3 — a b2.

Ans. 2a3a2b+4 a b2 - (a3 + b2 + a b2).

25. To what is 5a4b+3c + (→ 3a+2b-c) equal? Ans. 2a 2b2c.

26. To what is 6x2+2y2- (3 x2 + y2) — [2x2+4y2 -(4x2 y2)] equal?

Ans. 5x24y2.

2y2

61. When dissimilar terms have a common factor, the

subtraction may be indicated by prefixing the difference of their coefficients, inclosed in a parenthesis, to the common factor.

28. From ay

b take dy

- C.

Ans. (ad) y―b+c.

29. From a x + dy take bx3 — cy2.

Ans. (ab) x2 + (d+ c) y2.

30. From ax- bxcx take x + ax

bx.

Ans. (c-1) x.

d take bx2+ ex 2 d. Ans. ax2 bx2 + (c − e) x + d.

32. Take 3bxy — (5 + c) z2

(3a2b) (x + y) $

from (4a5b) √x + y + (a — b) x y — c z2.

Ans. (7a7b) √x + y + (a — 4 b) xy + 5 x2.

MULTIPLICATION.

62. MULTIPLICATION, in Algebra, is the process of taking one quantity as many times as there are units in another quantity.

The Multiplicand is the quantity to be multiplied, or

taken.

The Multiplier is the quantity by which we multiply. The Product is the result of the operation.

The multiplicand and multiplier are often called factors. The PRODUCT of factors is the same, in whatever order they are taken.

63.

For, the product contains one factor as many times as there are units in the other. Thus, the product of a × b, or ba, will be ab units, since b taken a times is the same as a taken b times. Let a 4 and b 4 X 3, or 3 × 4, equal to 12.

3, we have