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, 3 a-2b+c is homogeneous of the first degree, 8 a3b-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 identical.

Reduction of Polynomials.

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

tive terms.

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

EXAMPLES.

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

Ans. 13 a4.

2. Reduce the polynomial 5 a 6 + 3 / a b2c7ab+ 17 ab+2ab2 c—6 a b — 8ab2c-10 ab +9 ab to its simplest form. Ans. 8a4b3ab2 c.

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

Ans. 0.

Addition.

SECTION II.

Addition.

21. Addition consists in finding the quantity equivalent to the aggregate or sum of several different quantities.

22. 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.

-

7. Find the sum of 6f, 9f, 13f, and -8f.

b.

Ans. 8f.

36.

-4b, and 136. Ans.

9. Find the sum of √x+ax — ab, ab −√x+xy,

ax + xy-4 ab, √x+√x− x and xy + xy+ax.

Ans. 2+3 ax-4ab+4xy-x.

Subtraction.

10. Find the sum of 7 x26x+5x3 z + 3-g

23. Subtraction consists in finding the difference between two quantities.

24. 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 negative terms; then AB is the quantity to be subtracted, and let C denote the quantity from which it it is to be taken.

If A alone be taken from C, the remainder CA 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— Á 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 C the quantity A — B, with its signs reversed, so as to make it -AB. Hence,

To subtract one quantity from another, change the signs of the quantity to be subtracted from + to —,

Multiplication of Monomials.

and from to +, and add it with its signs thus reversed to the quantity from which it is to be taken.

9. From 3 a 176-106+13a-2 a

-

take 66-8a-b-2a+3d9a-5h.

Ans. 15 a 32b-3d+5h.

-

10. Reduce 32 a+3b-(5a+176) to its simplest

form.

Ans. 27 a - 14 b.

Ans. 7a-5 b.

11. Reduce a+b-(2a-3b) — (5a+7b) —

(-13 a +2b) to its simplest form.

SECTION IV.

Multiplication.

25. Problem. To find the continued product of several monomials.

Solution. The required product is indicated by writing the given monomials after each other with the sign of multiplication between them, and thus a monomial is formed, which is the continued product of all the factors of the given