55. Order of Operations. In a chain of operations involving the signs, +, −, ×, ÷, the numbers connected by the signs > and must be operated upon first from left to right in the order in which they occur. The results thus obtained should

be added and subtracted as indicated by the signs + and -.

2.

7-(-6)+(−12).

3.

3x(-4)+(− 5)— 70.

4. 0-2x(-3)+7 − (− 1).

5. -1x(-2)x (− 3)+6 × (− 2).

6.5+3x7 −(−5) × (−4).

18. 15÷(5) + 8 × (− 2) — 7 + (− 3).

19. 12 × (2) + (−3) × 8 - 10 × (− 1).

}

20. 0-10 x (-2)-(-4) × 8-7-(-7).

21. 1215 (-13) - 15-(- 15).

22. 12 × (− 1)− (− 10) × (− 1) — 8.x (− 1) — (— 6)(− 1).

=

=

If a = 6, b = − 5, c — — 3, d — — 1, e = — }, find the values of the expressions in examples 23 to 43.

46. Does x =— 1 satisfy the equation x2 - 2 x − 3 = 0 ?

REVIEW EXERCISE

57. 1. What quality signs would you associate with each of the following: north latitude, south latitude? rising temperature, falling temperature? debts, credits, money lost, money spent, money earned, money found? A.D., B.C.? points won in a game, points lost, penalties?

2. Compare the addition of a negative number with the subtraction of a positive number having the same absolute value. Illustrate.

3. Indicate the net result of $10 earned, $3 spent, $2 found, and $2 spent.

4. The temperature at 8 o'clock was 28°; at 10 o'clock it had risen 4°; at noon it was 5° warmer than at 10 o'clock; at 2 o'clock it had risen 2° more; at 4 o'clock it was 3° colder than at 2 o'clock; at 6 it was 4° below the temperature at 4 o'clock; and at 8 P.M. it was 7° colder than at 6 o'clock. (1) Indicate by arithmetical additions and subtractions the temperature at 8 P.M. (2) Find the same result by addition of signed numbers.

5. If you walk 3 miles south and 7 miles north, how far and in what direction from the starting point are you? Indicate by adding signed numbers.

6. How far upstream are you if you have rowed 7 miles up and drifted 2 miles down? Indicate the process of finding the answer in two ways.

7. Pikes Peak is 14,108 feet above sea level. A place in Holland is 16 feet below sea level. How much higher is Pikes Peak than the place in Holland? Indicate two ways of finding the answer.

8. If a gasoline launch can run 14 miles an hour in still water, how fast can it run up a river whose current flows 4 miles an hour? How fast can it run downstream?

9. If a person can swim 21 miles an hour in still water, represent his rate when swimming against a current of 3 miles an hour. Represent his rate downstream.

10. The Roman Empire fell 476 A.D., 622 years after the fall of Carthage. What was the date of the fall of Carthage? 11. Give the rules for addition, subtraction, multiplication, and division of signed numbers.

12. Define subtraction; define division.

13. What is the basis of the rule for subtraction of signed numbers? of the rule for division?

14. What is the absolute value of a number?

15. What is the sign of (-1)10? of (-1)"? Can you give an answer that will apply to all such examples?

16. What is the "order of operations"?

III. ADDITION

58. Algebraic Expression. A number represented by algebraic symbols is an algebraic expression.

Thus, 2 ab, 5 - 3 ab, 4 + 2 b are algebraic expressions.

is a

59. Monomial, Term. An algebraic expression the parts of which are not separated by either of the signs + or -monomial or a term.

An algebraic expression consisting of

two or more terms is a polynomial.

Thus, 3 ax 4 c + 7 and m − n + 11 xy — 16 are polynomials.

The monomials that make up the polynomial are the terms of the polynomial.

Thus, 3 ax,

4 c, and 7 are the terms of the polynomial 3 ax

4c+7.

A polynomial of two terms is a binomial, and one of three terms is a trinomial.

Thus, 2 a b is a binomial, and ax

by+c is a trinomial.

ORAL EXERCISE

61. In the following expressions, name (a) the monomials, (b) the binomials, (c) the trinomials, (d) the polynomials :