12

Ι

7. Add 4ax 130 + 3x2, 5x2 + 3xx -† 9x2, 7xy 90, and √x + 40 6x2 together. 8. Add 2a2 3ab +263 3a2, 363 2b35ab + 100, and 20ab + 16a2 Ans. 13a+22ab + 3b3 + a3 3c2 7 ✓ bc ab + x.

5α

+

9

) and +

b

d

9. Add

a

ac

+6 (ab +*) together.

5a2 26 2a2 + 5b 3a2 76

13a

Ans.

b

a

XC

10. Add 3a2 + 4bc e2 + 10, 5a2 + 6bc + 2e2 — 15, and 4a2 9bc — 10e2 + 21 together.

Ans. bc 6a2 9e2 + 16.

3x2

+

I

·4x2 +

Ans. 7ax +8x2 + 7xy.

EXAMPLES.

x2

2y + 3 4x2 + 9y

5

1ly +8

a

4c2

SUBTRACTION.

SUBTRACTION is the taking of one quantity from another or the method of finding the difference between any two quantities of the same kind; which is performed as follows:

*

RULE.-Change all the signs (+ and -) of the lower line, or quantities that are to be subtracted, into the contrary signs, or rather conceive them to be so changed, and then collect the terms together, as in the several cases of addition.

•

c+b-b

5

2a2 + a3 — 5c3, 4c3 80 together.

be

c3 +20 - bc.

8a

762

b

a

* This rule being the reverse of addition, the method of operation must be so likewise. It depends upon this principle, that to subtract an affirmative quantity from an affirmative, is the same as to add a negative quantity to an affirmative.

Thus, according to Laplace, we can write

a = a + b — b

5xy + 8x 8x 3xy

2xy +16x+5

a

C =

•

so that if from a we are to subtract +b, or the same thing, if in a we suppress + b, or transformation (1), must be ab in the first case, and a+b in the second. Also, if from ac we take away +b, orb, the remainder, from (2), will be a-c-b, or a-cb.-ED.

૩

(1),
(2);

; or what amounts to

b; the remainder from

I

7x3

8y2y5 -y2+3y+2

9y2 — 5y — 7

4√ ax 2x2y

5xy2

3. From 3a + b + c — 2d, take b

C

EXAMPLES FOR PRACTICE.

2

1. Find the difference of (a + b) and 1⁄2 (a - b). Ans. b. 2. From 3x 2a b+7, take 8

ac

3b+ a + 4x. Ans. 2b 8c + 2d 8. Ans. 3a9c 4. From 13x2 2ax+962, take 5x2- 7ax b2.

3 √ ax

ax-2xy +5xy2x2+8x+2√x-4y

5x2 + √x-4y

6x2 8x

3xy

x + 3―xy

1

a.

5. From 20ạx − 5 √ x + 3a, take 4ax + 5x2 Ans. 16ax 10 ✓ x + 4α. 6. From 5ab + 2b2 — c + bc — b, take b2 — Qab + bc. Ans. 7ab+b2 c — b

7x -2xy

7. From ax3 bx2 + cx d, take bx2 + ex 2d. Ans. ax3 2bx + (c − e) x + d. 8. From - 6a — 4b — 12c + 13x, take 4x 9a + 4b 5c. Ans. 3a +9x 86 - 7c.

Amanda

BERNANCE

Ans. 8x2+5ax + 106a.

I

x2

8ax — 70, and 30 — 4 √ (x) — 2x2 + 4a2x2. I

3a - 1.

4d + 8.

9. From 6x3y — 3√(xy) — 6ay, take 3x2y + 3 (xy)3 — 4ay. Ans. 3x2y — 6 √(xy) — Qay.

I

10. From the sum of 4ax- 150+ 4x3, 5x2 + 3ax +10x2, and 90 2ax 12 √(x); take the sum of 2ax — 80 + 7x3,

Ans. 11ax+ 60 — x2 4a2x2

MULTIPLICATION.

MULTIPLICATION, or the finding of the product of two or more quantities, is performed in the same manner as in arithmetic; except that it is usual, in this case, to begin the operation at the lefthand, and to proceed towards the right, or contrary to the way of multiplying numbers.

The rule is commonly divided into three cases; in each of which it is necessary to observe, that like signs, in multiplying, produce+, and unlike signs, -.

It is likewise to be remarked, that powers, or roots of the same quantity, are multiplied together by adding their indices: thus,

I

a × a3, or a1 × a2 = a3; aa × a3 = a3; aa × a3 = a; and

am X an = am+n.

The multiplication of compound quantities, is also, sometimes, barely denoted by writing them down, with their proper signs, under a vinculum, without performing the whole operation, as

3ab (a — b), or 2a √ (a2 + b2). Which method is often preferable to that of executing the entire process, particularly when the product of two or more factors is to be divided by some other quantity, because, in this case, any quantity that is common to both the divisor and dividend, may be more readily suppressed; as will be evident from various instances in the following part of the work.*

CASE I.

When the factors are both simple quantities.

RULE.-Multiply the coefficients of the two terms together, and to the product annex all the letters, or their powers, belonging to each, after the manner of a word; and the result, with the proper sign prefixed, will be the product required.†

*The above rule for the signs may be proved thus: If B, b, be any two quantities, of which B is the greater, and B b is to be multiplied by a, it is plain that the product, in this case, must be less than ав, bеcause B -b is less than B; and, consequently, when each of the terms of the former are multiplied by a, as above, the result will be (B—b) × α = аB --a

ab.

For if it were aв+ab, the product would be greater than aв, which is absurd.

Also, if в be greater than b, and a greater than a, and it is required to multiply B- -b by a a the result will be

ав

K

(Bb)X(sa) — AB — аB — b▲ † ab. For the product of B b by A is A (Bb), or AB- ·Ab, and that of B-b by b bya, which is to be taken from the former, is a (в - b) as has been already shown; whence в - b being less than в, it is evident that the part, which is to be taken away must be less than aв; and consequently since the first part of this product is aв, the second part must be ab; for if it were -ab, a greater part than aв would be to be taken from A (B --b), which is absurd.

B

w

+ When any number of quantities are to be multiplied together, it is the same thing in whatever order they are placed: thus, if ab is to be

За 26 4a

12a2 -8ab

12x · ab

Catal

5a

- ab.

1

13x2
2a

26αx2+2a3b

+ 5a

6x

EXAMPLES.

6xy

3x

CASE II.

When one of the factors is a compound quantity. RULE.-Multiply every term of the compound factor, considered as a multiplicand, separately, by the multiplier, as in the former case; then these products, placed one after another with their proper signs, will be the whole product required.

8

18x3y —24x

35x-7a

2x

-70x2 + 14ax

25xy + 3a2

13x2

325x3y + 39a2x2

3y2+y-2 xy

3xy3 + xy3 —2xy

3x2-xy + 2y2

5x2

15x1-5x3y-10x2y2

multiplied by, the product is either abc, acb, or bca, &c.; though it is usual, in this case, as well as in addition and subtraction, to put them according to their rank in the alphabet. It may here also be observed in conformity to the rule given above for the signs, that (+-a) ×(-+-6). or (——a) × (--b) =+ab; and ( + a) × ( —b), or ( − a) × (+

CASE III.

When both the factors are compound quantities. RULE.-Multiply every term of the multiplicand separately, by each term of the multiplier, setting down the products one after another, with their proper signs; then add the several lines of products together, and their sum will be the whole product required.

10xy — 8y2

15x2+2xy — 8y2

x2 + y

x2 + y

2

x2 + x2y
+ x3y + y2

x2 + 2x2y + y2

x2 + xy — y2

X - Y

x3 + x3y — xy2

X3

3. Required the product of a2 + xy + y2 Ans.

x2y — xy2 + y3

*

x2 + xy + y2

30 -y

x2 + x2y + xy2

EXAMPLES FOR PRACTICE.

1. Required the product of x2- xy + y2

and x+y. Ans. 3y3. 2. Required the product of x3 + x2y + xy2 + y3 and x

B

y.

Ans x - y1.

4

and x2-xy -+ y2. xxy3 +ya.

2

.3

2xy2+ y3

4. Required the product of 3x2 - 2xy +5, Ans. 3x+4x3y — 4x2y2

and x2+2xy — 3. 4x2 + 16xy

- 15.

5. Required the product of 2a2 3ax 4x2 and 5a2 6ax 202. Ans. 10a1 27a3x + 34a2x2 —18ax3 8x1. 6. Required the product of 53+ 4x2 + 3a2x + a3, and 3ax + a2. Ans. 10x5 7ax1 a2x3 3a3x2+a3. 7. Required the product of 33 + 2x2y2+ 3y3 and 2x33x2y2 +5y3.

2x2

4.4

dx

Ans. 6x6 5x5у2-6x11+21x3y3 + x2 + 15yo. 8. Required the product of a3 ax2 + bx c and x2 +e. Ans. a ax2- dx2 + (b + ad + e) x3- (c+bd+ae) — x2+(cd + eb) x-ce.

4

— y3