SECTION II.

ALGEBRAIC EXPRESSIONS.

26.-Ex. 1. If a express a quantity, what will express five times that quantity?

2. If a express one quantity and b another quantity, by what sign will you connect them to express their sum?

3. In the expression a+b, what are the parts connected? In a-b, what are the parts connected?

=

4. If a 8 and 6-3, what is the value expressed by a+b? By a-b?

Definitions.

27. An Algebraic Expression is a quantity expressed in algebraic language.

Thus, 5a is the algebraic expression of 5 times the quantity denoted by a.

28. The Terms of an algebraic expression are the parts connected by the sign + or−.

Thus, ab and cb are the terms of ab+cb, and x and y the terms of x-y.

Terms are positive or negative according as they have the sign + or -.

Thus, in ab-cb, the term ab is positive, and the term ative.

- cb is neg

29. Similar Terms are those which contain the same letters having the same exponents.

Thus, 2xy' and -7xy' are similar terms.

30. Dissimilar Terms are those which contain different letters

or different exponents of the same letter.

Thus, 4a2b and 3ac2 are dissimilar terms; also 4a2b and 4ab2.

31. A Monomial is an algebraic expression consisting of one term.

Thus, ab2, -bx, etc., are monomials.

32. A Polynomial is an algebraic expression consisting of more than one term.

Thus, a c and a+ab2 -b are polynomials.

33. A Binomial is a polynomial consisting of two terms.

Thus, ab2+cd and xy – 3x2y are binomials.

34. A Trinomial is a polynomial consisting of three terms. Thus, ax+xy-ab and cd+bx+b are trinomials.

35. The Degree of a term is the number of its literal factors. Thus, 2x, which contains only one literal factor, is of the first degree, and 6ab2, which contains three literal factors, is of the third degree.

36. The Numerical Value of an algebraic expression is the result obtained by substituting for its letters definite numerical values, and performing the processes denoted by the signs.

Thus, the numerical value of 6a+b2 - c, when a = 2, b=5 and c=11, is 6×2+5×5-11, which equals 26.

EXERCISES.

37. Express algebraically—

1. The sum of x and y.

2. The value of 2 times a diminished by b.

Ans. x+y.

3. Five times x, added to three times y, diminished by c times d. Ans. 5x+3y-cd.

4. The difference of a and c multiplied by the sum of a and b. Ans. (a-c) (a+b).

5. Two times a multiplied by b square, plus four times b multiplied by e cube. Ans. 2ab2+4bc3.

6. a third power into b fourth power, plus two times a second power into c, minus three a second power into b third power.

7. Three times x plus y, divided by seven times the product of a multiplied by b.

8. a minus b, divided by a plus b.

9. a minus b, multiplied by c into x.

10. The square root of (a-c).

3x+y

Ans.

7ad

Ans. (a - b)cx.

Ans. Va

C.

11. The cube root of a, plus the fourth root of x plus y.

12. Write a polynomial of three terms.

Ans. Va+x+y.

38. What are the numerical values of the following expressions, when a = 3, b = 5, c = 2, d = 7, m = 2, and n = 3?

10. If x=6 and y = 8, what is the value of 2x+(x+y)x+15?

Ans. 111.

11. If a = 15, b=10 and c=25, what is the value of √b2+1/4c-2a? Ans. - 10.

12. If x=11, y=9 and a=20, what is the value of (x+y) (x-y)+1/5a?

13. If a = 5, b=6, m=4 and n = 10, what is the value of 10a+by 1000n3-v/m3 ? Ans. 642.

14. If a = 4, b = 10, x = 12 and y = 16, what is the value of (a+b)(ab+8)+v/x3 − 2y?

3

SECTION III.

ALGEBRAIC PROCESSES.

39. A Problem is something to be done or a question to be solved.

40. Algebraic Processes are the means employed in the solution of problems whose conditions can be expressed in algebraic language.

PROBLEMS.

41.-Ex. 1. Henry has twice as many books as Arthur, and both together have 24. How many books has each ?

SOLUTION. Let x equal Arthur's number, then 2 times x, or 2x, will equal Henry's number, and 3x will equal the number both together have, which is 24.

If 3x equal 24, x, or the number Arthur has, must equal one-third of

x= Arthur's number; 2x Henry's number. x+2x=24

3x=24

x= 8, Arthur's number. 2x=16, Henry's number.

24, or 8, and 2x, the number Henry has, must equal 2 times 8, or 16.

2. A man is four times as old as his son, and the sum of their How old is each of them?

ages is 55 years.

Ans. The man, 44 years; the son, 11 years.

3. A farmer gave for a farm and its stock $3000, and the farm cost 5 times as much as its stock. What was the cost of each? Ans. The farm, $2500; its stock, $500.

4. The sum of two numbers is 200, and the larger is 3 times the smaller. What is the larger number?

5. My salary is $3000 a year, and the portion of it I spend is 4 times the portion I save. What portion of it do I spend?

6. My horse and carriage are together worth carriage is worth twice as much as the horse. value of each?

Ans. $2400.

$750, and the

What is the

42.-Ex. 1. A plow, a harrow and a cart cost together $96. The harrow cost twice as much as the plow, and the cart 5 What did each cost?

times as much as the plow.

SOLUTION. Let x equal the cost of the plow; then 2x will equal the cost of the harrow, 5x the cost of the cart, and 8x the cost of all together, which is $96.

If 8x equal $96, x, or the

cost of the plow, must equal

one-eighth of $96, or $12;

2x, or the cost of the harrow, must equal 2 times $12, or $24; and 5x, or the cost of the cart, must equal 5 times $12, or $60.

2. The sum of three numbers is 180; the second is 5 times the first, and the third 6 times the first. bers?

What are the num

Ans. 15, 75, 90.

3. Among three men, A, B and C, $2000 were distributed, so that B and C each had twice as many dollars as A. How many dollars did each have?

4. The sum of the ages of three persons, A, B and C, is 96 years. B is twice as old as A, and C 3 times as old. How old are B and C each ? Ans. B, 32 years; C, 48 years.

5. Divide $600 among Smith, Downer and Herr, so that Downer shall have 3 times as much as Smith, and Herr twice as much as Downer. What will be the share of each?

6. A man has three farms, which together contain 320 acres, and the farms are in relative size as the numbers 1, 2 and 5. How many acres are there in each? Ans. 40, 80, 200

7. John has some marbles, George has four times as many as John, William has twice as many as John and George, and they all have 150 marbles. How many has each?

8. Andrew has 5 times as many cents as Joseph, and Edwin has one-half as many as Andrew and Joseph. If the three have in all 135 cents, how many has each?