14. An algebraic expression made up of several terms is called a Polynomial, meaning many-termed.

15. A polynomial of two terms is called a Binomial, and one of three terms a Trinomial.

Thus, 2 a 5 d is a binomial; 3 m + 4 n − 2 a, a trinomial.

16. In the expression abc, the a, the b, and the c are called the Factors of the expression. An expression is a Multiple of any one of its factors, or of the product of any two or more of them.

In the expression 2 ac +3 bc, point out the terms; also the factors. How are factors distinguished from terms?

17. In a product any factor, or product of factors, is called the Coefficient of the rest of the product. In practice, the word is usually applied only to some factor whose numerical value is known or expressed, and which appears first in the product. Cf. Art. 100.

Thus, in the expression 2 abc, 2 is the coefficient of abc, 2 a of bc, etc. Since a = 1 a, the coefficient 1 is understood before any letter when no coefficient is expressed.

18. The product of two or more equal factors is called a Power. The product of two equal factors is called the second power; of three equal factors, the third power; and so on.

19. The number symbol that shows how many times the equal factor enters into a power is called an Exponent.

Thus, a2 is read "a square,' axa; a3 is read "a cube," or

66

or .6

a to the second power," and means a to the third power," and means a × a

xa; at is read "a to the fourth power"; and so on.

How is the exponent distinguished from the coefficient? In the expression 3 ab, of what is 3 the coefficient? What is the exponent of b?

Since may be regarded as taken once as a factor to make itself, b is regarded as meaning b1. Any letter is considered as having the exponent 1 when no exponent is expressed.

20. We have learned in arithmetic that one of the equal factors of a number is called a Root of the number. The square root is one of the two equal factors; the cube root is one of the three equal factors; and so on.

In expressions like “the square root of 3" we shall see that the word root has a broader meaning than that given in the ordinary definition, since 3 has not turo equal integral factors.

21. As in arithmetic, evolution is indicated by the symbol √, said to have had its origin in r, the initial letter of radix (Latin, root); but it will be found that both involution and evolution are more frequently indicated by exponents.

EXERCISES

22. 1. Write the expression which means that a is to be multiplied by the sum of 5 and 2.

2. Indicate the multiplication of y by the difference between 7 and 4.

3. Is the expression 5 ab + 10 be-5 be a polynomial? What other name may be given to it? What factors are found in each term ?

ах.

4. Write an expression having four terms and the factor

5. Write a trinomial containing the factor 3b.

6. Name the terms in the expression 6 a 2 (a - b) + cd.

7. Write a monomial whose factors are 2, a, x, and 3.

8. Write an expression which is a multiple of 5, b, c, 2, Is it a monomial? Why?

and y.

9. Write an algebraic expression containing four signs of operation. How many terms does it contain?

10. Why is the expression, a 3(a+b), not a trinomial?

11. Write a monomial containing the factor a, the coefficient 5, and the exponent 2.

12. Write a binomial, each term containing the exponent 3 and the coefficient 4.

13. Use symbols to express the following:

Four a square plus three times b square.

Five a cube plus twice the third power of b.

Ten a cube minus six times the sum of b and c.

Six a times y cube plus four times the product of b and y.

14. What is the coefficient of a in 2

3

? The exponent?

15. What is the meaning of the expression abc? Of 125? What is the value of abc, if a = 1, b = 2, c=5?

16. What is the value of each term in the expression, 2 a x

1

a, if a = 1, b = 2, c=3, d=4?

b÷dx c
of the polynomial?

THE EQUATION

What is the value

23. The Symbol of Equality, =, is read is equal to, and indicates that the two numbers or expressions between which it is placed are equal in value.

Thus, 235 means that the sum of 2 and 3 is equal to 5; and x= a means that x is equal to a, or has the value a.

24. 1. When will a scalebeam balance?

2. If a weight be added to one pan, what must be added to the other to preserve the balance?

3. If a weight be taken from one pan, what must be done to preserve the balance?

4. What does the expression 2x + 3: = 11 mean?

5. If we add 1 to the first side of the equality, how much must be added to the second side to make the sides equal again?

6. If we take 3 from the first side, what must be done to the second to preserve the equality?

7. Is there any difference in value between 7 +3 and 15 - 5? Then 7 + 3 = 15 – 5.

8. How much is 6 times (7+3)? How much is 6 times (155)? Are the products equal?

9. Then what may be done to both sides of an equality without destroying the equality?

10. What is the quotient of (7 + 3) ÷ 2? Of (15 — 5) ÷ 2 ? Are the quotients equal?

11. Then what else may be done to both sides without destroying the equality?

12. A 10-acre field is worth $100 an acre, and a 20-acre field is worth $50 an acre. In what respect are the two fields equal?

25. From the preceding exercises it will be seen that the two sides of any equality are like the weights in the two pans of a balanced scalebeam, and that, in general, any change made in one side requires a like change in the other if the equality is to be preserved.

26. When any two numbers or expressions are connected by the symbol of equality, the whole expression thus formed is broadly called an equation. This definition includes both the identity and the equation proper.

27. In the stricter sense in which the word is used in algebra, an Equation is an equality that exists only for particular values of the letters representing the unknown numbers. These values are called the roots of the equation.

Thus, 3x+2

=

8 is an equation because the equality exists only when r has the value 2. If x=3, there is no equality.

28. An Identity is an equality that exists for any value that may be substituted for the letters involved.

Thus, 9 a + 3 a = (3 × 4) a is an identity because the equality exists for any value that a may have. Any equality that involves numerals only is an identity, as 2 + 3 = 5.

1. An equation states a condition, and the values of the unknown numbers are to be found. An identity states that one of two expressions is equal to the other, or can be reduced to the other; it is to be proved. When the roots of an equation are substituted for the letters representing its unknown numbers, the equation reduces to an identity.

2. An identity is indicated by the symbol; as, a (a —b) = a2 — ab. It is used in this book in all cases where it is desired to emphasize the fact that any particular equality is an identity.

29. The part of an equation that is written before the sign is called the first member, and that written after the sign, the second member.

Thus, in the equation x + a = 10, x + a is the first member. What is the second member?

The two members may differ widely in form, but in value they must be the same.

30. Finding the values of the unknown numbers - the roots of an equation is called solving the equation. If the value found for the unknown number is substituted in the original equation, and the equation reduces to an identity, the value of this unknown number (called the root of the equation) is said to be verified. This substitution must always be made in the original equation.

31. The extensive use of the equation is the most characteristic feature of algebra; it makes possible the solution of practical problems otherwise insolvable.

32. The Symbols of Inequality are >, meaning is greater than, and <, meaning is less than. The symbols, not equal to; K, not greater than; and, not less than, also, are used in algebra.

"; cd is read "c is less

Thus, ab is read "a is greater than b" than d"; etc.