Signs of Addition, Subtraction, Multiplication, and Division.

3. The sign is called minus or less, or the negative sign, and placed between two quantities denotes that the quantity which follows it is to be subtracted from the one which precedes it.

Thus 7- - 2 is 7 minus or less 2, and denotes the remainder after subtracting 2 from 7. Likewise a - b is the remainder after subtracting b from a.

4. The sign X is called the sign of multiplication, and placed between two quantities denotes that they are to be multiplied together. A point is often used instead of this sign, or, when the quantities to be multiplied together are represented by letters, the sign may be altogether omitted.

Thus 3 × 5 × 7, or 3.5.7 is the continued product of 3, 5, and 7. Likewise 12 X a × b, or 12. a. b, or 12 ab, is the continued product of 12, a, and b.

5. The factor of a product is sometimes called its coefficient, and the numerical factor is called the numerical coefficient. When no coefficient is written, the coefficient may be considered to be unity.

Thus in the expression 15 a b, 15 is the numerical coefficient of a b; and in the expression xy, 1 may be regarded as the coefficient of xy.

6. The continued product of a quantity multiplied repeatedly by itself, is called the power of the quantity; and the number of times, which the quantity is taken as a factor, is called the exponent of the power.

The power is expressed by writing the quantity.

Coefficient. Power. Root.

once with the exponent to the right of the quanity, and a little above it. When no exponent is written, the exponent may be considered to be unity.

Thus the fifth power of a is written a5, and the exponent of a may be regarded as 1.

7. The root of a quantity is the quantity which, multiplied a certain number of times by itself, produces the given quantity; and the index of the root is the number of times which the root is contained as a factor in the given quantity.

The sign is called the radical sign, and when prefixed to a quantity indicates that its root is to be extracted, the index of the root being placed to the left of the sign and a little above it. The index 2 is generally omitted, and the radical sign, without any index, is regarded as indicating the second or square

root.

Thus, a ora is the square root of a,

8. The signs

and are called the signs of division, and either of them placed between two quantities denotes that the quantity which precedes it is to be divided by the one which follows it. The process of division is also indicated by placing the dividend over the divisor with a line between them.

denotes the quotient of a di

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 a> b 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 the sign.

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.

is

13. An algebraic expression composed of several terms, connected together by the signs and —, called a polynomial, one of two terms is called a binomial, one of three a trinomial, &c.

14. The value of a polynomial is evidently not affected by changing the order of its terms.

Thus, α- -b-cd is the same as a c a+d-b-c, or b+d+a—c, &c.

b+d, or

Degree, Dimension, Vinculum, Bar, Parenthesis, Similar Terms.

15. 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 exponents of its literal factors.

Thus, 7 x is of one dimension, or of the first degree; 5a2bc 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, 3a-2b+c is homogeneous of the first degree, 8a3b-16 a2 b2b4 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) x d is the product of a+b+c by d, √x2+y2, or √(x2 + y2) is the square root of x2 + y2. The bar is sometimes placed vertically.

(a−2b+3c) x+(5a2+3—2d) x2+(−3c+4d—1)x3.

18. Similar terms are those in which the numerical coefficients being neglected, the literal factors are

Reduction of Polynomials.

19. The terms of a polynomial which are preceded by the sign are called the positive terms, and those which are preceded by the sign

tive terms.

are called the nega

When the first term is not preceded by any sign it is to be regarded as positive.

20. The following rule for reducing polynomials, which contain similar terms, is too obvious to require demonstration.

Find the sum of the similar positive terms by adding their coefficients, and in the same way the sum of their similar negative terms. The difference of these sums preceded by the sign of the greater, may be substituted as a single term for the terms from which it is obtained.

When these sums are equal, they cancel each other, and the corresponding terms are to be omitted.

Thus, a2b-9ab2+8a2 b+5c-3a2b+8 a b2 — 2a2b+c+ab2. -8 c is the same as 8 a2 b

2 c.

EXAMPLES.

1. Reduce the polynomial 10 a1 +3 a1 +6 a1 — a1 — 5 a to its implest form.

Ans. 13 a*.

2. Reduce the polynomial 5 a b + 3 / a b2c - 7ab+ 17 ab+2ab2 c — 6 a1 b — 8ab2c-10 ab +9 ab to its simplest form. Ans. 8a4b-3ab2 c.

3. Reduce the polynomial 3 a-2 a−7ƒ+3ƒ+2 a +4f-3 a to its simplest form.

Ans, 0.