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Signs of Equality and Inequality. Algebraic Quantity.
9. The sign = is called equal to, and placed between two quantities denotes that they are equal to each other, and the expression in which this sign occurs is called an equation.
Thus, the equation a=b denotes that a is equal to b.
10. The sign > is called greater than, and the sign < is called less than; and the expression in which either of these signs occurs is called an inequality.
Thus, the inequality ab denotes that a is greater than b; and the inequality a < b denotes that a is less than b; the greater quantity being always placed at the opening of
11. An algebraic quantity is any quantity written in algebraic language.
12. An algebraic quantity, in which the letters are not separated by the signs + and is called a monomial, or a quantity composed of a single term, or simply a term.
Thus, 3 a?, 10 a? x are monomials.
13. An algebraic expression composed of several terms, connected together by the signs + and is called a polynomial, one of two terms is called a binomial, one of three a trinomial, &c. Thus, a? + b is a binomial,
c+2 - y is a trinomial, &c. 14. The value of a polynomial is evidently not affected by changing the order of its terms.
Thus, a-b-c+d is the same as a -6-6+d, or atd-6-c, or-6+d+a-c, &c.
Degree, Dimension, Vinculum, Bar, Parenthesis, Similar Terms.
16. Each literal factor of a term is called a dimension, and the degree of a term is the number of its dimensions.
The degree of a term is, therefore, found by taking the sum of the erponents of its literal factors.
Thus, 7x is of one dimension, or of the first degree ; 5 a b c is of four dimensions, or of the fourth degree, &c.
16. A polynomial is homogeneous, when all its terms are of the same degree.
Thus, 32-26+c is homogeneous of the first degree. 8a8b-16 ao 12 + 14 is homogeneous of the fourth degree.
17. A vinculum or bar -, placed over a quantity, or a parenthesis ( ) enclosing it, is used to express that all the terms of the quantity are to be considered together.
Thus, (a+b+c) xd is the product of a + b + c by d, ✓22 +y?, or ✓ (x2 + y2) is the square root of x2 + y2. The bar is sometimes placed vertically. Thus,
u +5 a? | 22 23 - 26
+30 2 d is the same as (a-26+3c)2+(5 a?+3-2d)+(-3c74d-1)+".
18. Similar terms are those in which the literal factors are identical.
7 ab and 3 a b are similar terms,
5 a4 b8 and 3 a4 b8 are similar ; 2 at 73 and 2 a3 64 are not similar.
a polyet? hit incerte 1 by the sig
Lerns, and which are .
are called negative,
When blene farm is not preceded by any ani is to be re jus puscilet.
20. TI to!!f ing riile fi: reducing poisi omials, which co ii. sim. ir terms, is too obvlu's t'y require demonstr
Find ile son of the sinitur positive te ms by rzut in ing their conthriant. , and in the serve 2114 ther" in of the similadi rative terms. The affererefons of these sur
scedil by the sign of te riuts; muy be substit' s as si single term for th term from which it igiturteork's
Wher thrza $**" are inuul they cancel, F.1 ch o her; and the risponi interns are to be omiied.
Thus, - . -- 9 ab? + 8? 19... +50-- 3 az 1, 56 a 627 2 a2 b +
ta!2 Sc is the ame as 8 a b ---20.
1. Reduce the polynomas i 9 ai + 3 tua
+5 af to its simplest form.
A is. 13 4 2. Reduce the polyactriai 562 V012. Tabt 17 ab+%valca* b-Eva'--- 10 ab- 946 to its simplest form,
B3 124. 3. Reance the peynumul 3 a - '0-78-38 +2 +4f- a to its sir ;lest form.
22. Addition consists in finding the quantity equivalent to the aggregate or sum of several different quantities.
23. Problem. To find the sum of any given quantities.
Solution. The following solution requires no demonstration.
The quantities to be added are to be written after each other with the proper sign between them, and the polynomial thus obtained can be reduced to its simplest form by art. 20.
1. Find the sum of a and a.
Ans. 2 a. 2. Find the sum of 11 x and 9 x.
Ans. 20 x. 3. Find the sum of 11 x and
Ans. 2 2. 4. Find the sum of 11 x and 9 2.
Ans. 5. Find the sum of 11 x and
Ans. 6. Find the sum of a and -b.
-b. 7. Find the sum of - 69,9f, 13f, and -8f. Ans. 8f. 8. Find the sum of — 126, — 46 and 13b. Ans. — 36.
9. Find the sum of Vitar- ab, ab-vitay, ar+xy— 4 ab, vitve - 1 and xy trytar.
Ans. 2 Ve +3 ar-4ab+ 4xy-*.
10. Find the sum of 7 x? - 6 vit57z+3-8 -223-VT
-8-8 - 22+ VI-3rz-1+7g -212 +3V++3x z-]-g
22 +8 VT-528 % +9-g Ans. 422 + 3 VT +2+5g.
25. Subtraction consists in finding the difference between two quantities.
26. Problem. To subtract one quantity from another.
Solution. Let A denote the aggregate of all the positive terms of the quantity to be subtracted, and B the aggregate of all its n' itive terms; then A - B is the quantity to be subtracted, .nd let C denote the quantity from which it is to be taken.
If A alone be taken from C, the remainder C-A is as much too small as the quantity subtracted is too large, that is, as much as A is larger than A – B. The required remainder is, consequently, obtained by increasing C- A by the excess of A above A- B, that is, by B, and it is thus found to be C - A+B.
The same result would be obtained by adding to the quantity A-B, with its signs reversed, so as to make it - A+B. Hence,
To subtract one quantity from another, change the signs of the quantity to be subtracted from + to,