(d) When several quantities are to be added together, it is the same thing, in whatever order they are placed. Thus, a + b-c, − c + a + b, or a-c+ b, are equivalent expressions; though it is usual, in such cases, to make the leading term positive.

Note. When quantities with literal coefficients are to be added together, it may be done by placing the coefficients, with their proper signs, under a vinculum, or between brackets, and then joining the sum or difference, thus arising, to the common quantity, as below:

ax + b cx + d

ax2 + bx cx2-dr

(a+c)x+b+d

(a + c)x2 + (b − d)x

EXAMPLES FOR PRACTICE.

1. It is required to add 4(a+b) and ÷(a−b) together.

a

2. Add 5x-3a+b+7 and - A a − 3x + 2b - 9 together.

3. Add 2a+3b-4c-9 and 5a-3b+2c-10 together.

4. Add 3a+2b − 5, a + 5b — c, and 6a − 2c + 3 together.

5. Add x ax + bx +2 and + cx2+ dx-1 together.

SUBTRACTION.

(c) 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: (e)

(e) The term subtraction, used for this rule, is liable to the same objection as that for addition; the operations to be performed being of a mixed nature, like those of the former. It

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

Note. When the quantities that are to be subtracted have literal coefficients, the operation may be performed by placing the coefficients, with their

may also be observed, that, though the rule here given, for subtracting one quantity from another, is universally applicable, it will not always be necessary to have recourse to it; for if 3a is to be taken from 7a, it is plain that the remainder will be 4a, without any consideration of the change of signs.

proper signs, between brackets, as in addition, and then subjoining the common quantity: thus,

EXAMPLES FOR PRACTICE.

1. Let (a-b) be subtracted from (a+b). 2. From 3x-2a-b+7, take 8-3b+a+4x. 3. From 3a+b+c-2d, take b-8c+2d-8. 4. From 5ab+ 2b2 − c + bc − b, take b2 — 2ab+bc. 5. From ax3- bx2 + cx - d, take bx2 + ex — 2d.

MULTIPLICATION.

(D) 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 left hand, 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 also to be remarked, that powers, or roots of the same quantity, are multiplied together by adding their indices: thus,

a × a2, or a1 × a2 = a3; aa × a3 = a'; a* × a3 = a

a" × a" = a+n.

a&; and

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 (f).

(f) 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 multiplied by c, 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) × ( + b), or ( − a) × ( − b) ≈ + abj and (+a) × (-b), or (-a) × ( + b) — ub.

C

=