COMPOSITION OF ALGEBRAIC QUANTITIES.

22. An algebraic quantity may consist of a single letter o element, or a combination of symbols as factors, or several com. binations or parts. The parts are called terms; hence,

23. The Terms of an algebraic quantity are the parts or divisions made by the signs + and -. Thus, in the quantity 5a+2b2 - cx, there are three terms, of which 5a is the first, 263 the second, and cx the third.

24. When a quantity consists of a single term, it is said to be simple; when it is composed of two or more terms, it is said to be compound.

25. Positive Terms are those which have the plus sign; as +x, or +2c2d. The first term of an algebraic quantity, if written without any sign, is positive, the plus sign being understood.

26. Negative Terms are those which have the minus sign; as -3a, or —2mx2. The sign of a negative quantity is never omitted. 27. A Coefficient is a number or quantity prefixed to another quantity, to denote how many times the latter is taken. Thus, in 3x, the number 3 is the coefficient of x, and indicates that x is taken 3 times; hence, the expression 32 is equivalent to x+x+x. In 4ax, 4 may be regarded as the coefficient of ax, or 4a as the coefficient of x. In 5 (a + x), 5 is the coefficient of a +x. When no coefficient is written, the unit 1 is understood.

28. It should be observed that in a term having the plus sign, the coefficient shows how many times the quantity is to be added; and in a term having the minus sign, the coefficient shows how many times the quantity is to be subtracted. Thus,

29. Similar Terms are terms containing the same letters, affected with the same exponents; the signs and coefficients may differ, and the terms still be similar. Thus, 3x2 and 72 are similar terms; also, 2md2 and 5md2 are similar terms.

30. Dissimilar Terms are those which have different letters or exponents. Thus, axy and ayz are dissimilar; also 3x2y and 3x3y2.

31. A Monomial is an algebraic quantity consisting of only one term; as 3x, or

7xy.

32. A Polynomial is an algebraic quantity consisting of more than one term; as x + y, or 4a2 3x + m.

33. A Binomial is a polynomial of two terms; as a + b, or 3x 2.

34. A Residual is a binomial, the two terms of which are connected by the minus sign; as a b, or 4x-3y. 35. A Trinomial is a polynomial of three terms; as x+ y+z, or a · 3b2 + d.

36. The Degree of a term is the number of its literal factors. Since the exponents show how many times the different letters are taken as factors, the degree of a term is always found by adding the exponents of all the letters. Thus, a and 5y are terms of the first degree; a2 and 4ab are terms of the second degree ; x3, 3x2y, 3xy3, and 4xyz are terms of the third degree.

37. A Homogeneous Quantity is one whose terms are all of the same degree; as a3- 5x2y+ 3xyz.

38. A Function of a quantity is any expression containing that quantity. Thus az is a function of x; 3y2+ 2y 4 is a function of y.

AXIOMS.

39. An Axiom is a self-evident truth. The following axioms underlie the principles of all algebraic operations :

1. If the same quantity or equal quantities be added to equal quantities, the sums will be equal.

2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal.

3. If equal quantities be multiplied by the same, or equal quantities, the products will be equal.

4. If equal quantities be divided by the same, or equal quantities, the quotients will be equal.

5. If a quantity be both increased and diminished by another, its value will not be changed.

6. If a quantity be both multiplied and divided by another, its value will not be changed.

7. Quantities which are respectively equal to the same quantity, are equal to each other.

8. Like powers of equal quantities are equal.

9. Like roots of equal quantities are equal.

10. The whole of a quantity is greater than any of its parts. 11. The whole of a quantity is equal to the sum of all its parts.

EXERCISES IN ALGEBRAIC NOTATION.

40. In the examples which follow, it is required of the pupil simply to express given relations in algebraic language.

1. Give the algebraic expression for the square of a increased by 4 times b. Ans. a2 + 4b. 2. Give the algebraic expression for 7 times the product of x and y, diminished by 5 times the cube of z.

3. Indicate the quotient of 12 times the square of a minus 5 times the cube of b, divided by the sum of a and c.

4. If d represent a person's daily wages, what will represent his wages for 6 days? Ans. 6d.

5. An army drawn up in rectangular form, has 6 men in rank, and a men in file; of how many men is the army composed?

6. If a man labor m days in a week at c dollars per day, what will his earnings amount to in 7 weeks?

7. The length of a prism is a, the breadth c, and the altitude ac; required the solid contents. Ans. ac (ac).

8. A has 4m dollars, B has m times as many dollars as A, and C has 3 times as many dollars as B wanting d dollars; how many dollars has C?

9. A dealer sells b sheep and c calves, at an average price of m dollars per head; how much does he receive for all?

10. A man has 3 square lots measuring m rods on a side; how many acres in the three lots?

3m2

Ans.

160

11. From a rectangular piece of land whose length was a rods and whose width was b rods, there were sold c acres; how many acres remained unsold?

12. A ship laden with a barrels of flour, valued at m dollars

per barrel, met with a disaster by which 6 barrels were lost, and the remainder damaged to the amount of d dollars per barrel; what was the worth of the remainder? Ans. (a-b) (m—d).

13. A man. having c acres of land worth b dollars per acre, divided its value equally among m sons and one daughter; how many dollars did each receive?

14. A company of n persons began business with a joint capital of c dollars. The first year they gained b dollars, the second year they lost d dollars, the third year they doubled the capital with which they began that year, and then dissolved partnership, sharing equally their accumulated capital; what was each man's share?

COMPUTATION OF NUMERICAL VALUES.

41. The Numerical Value of an algebraic quantity is the number obtained by assigning numerical values to all the letters, and performing the operations indicated.

1. What is the numerical value of (a2 — bc) a, when a = 30, 25, and c = 28?

b

=

OPERATION.

(a2 = -be) a (30 x 30-25 x 28) x 30 200 x 30 = 6000, Ans.

Find the numerical values of the following expressions, in which a = 12; b = 10; c=8; m = 6; n = 5; d=2.

Find the numerical values of the following expressions, in whion

a=8; b=6; c=4; d=2; m = 3; n = 1.

SIGNIFICATION OF THE PLUS AND MINUS SIGNS.

42. The signs,+ and -, have been defined as symbols of operation, the former indicating addition, and the latter sub traction. Now when we meet with detached or single term affected with the plus or minus signs, as for instance in th examples of addition, subtraction, multiplication, or division