2. Write nine times the cube of a multiplied by b, diminished by the square of e multiplied by 'd.

Ans. 9a3b-c2d. 3. If a=2, b=3, and c=5, what will be the value of 3a2 multiplied by b2 diminished by a multiplied by b multiplied by c. We have

3a2b2-abc-3 × 22 × 32-2×3×5=78.

4. If a=4, b=6, c=7, d=8, what is the value of 9a2+bc-ad? Ans. 154. 5. If a=7, b=3, c=7, d=1, what is the value of Gad+3b2c-4d2? Ans. 227. 6. If a=5, b=6, c=6, d=5, what is the value of 9abc-8ad4bc? Ans. 1564. 7. Write ten times the square of a into the cube of b into e square into ď3.

17. When an algebraic quantity is not connected with any other, by the sign of addition or subtraction, it is called. a monomial, or a quantity composed of a single term, or simply, a term. Thus,

3a, 5a2, 7a3b2,

are monomials, or single terms.

18. An algebraic expression composed of two or more parts, separated by the sign+ or -; is called a polynomial, or quantity involving two or more terms. For example,

3a-56 and 2a2-3cb+4b2

are polynomials.

19. A polynomial composed of two terms, is called a binomial; and a polynomial of three terms is called a trinomial.

QUEST.-17. What is a monomial? Is 3ab a monomial? 18. What is a polynomial? Is 31-b a polynomial? 19. What is a binomial?

What is a trinomial?

20. Each of the literal factors which compose a term is called a dimension of this term: and the degree of a term is the number of these factors or dimensions. Thus,

3a}

5ab

7 a3bc2 = 7aaabce {

is a term of one dimension, or of the first degree.

is a term of two dimensions, or of the

second degree.

is of six dimensions, or of the sixth

degree.

21. A polynomial is said to be homogeneous, when all its terms are of the same degree. The polynomial

3a-2b+c is of the first degree and homogeneous.

-4ab+b is of the second degree and homogeneous. 5a2c-4c3+2c2d is of the third degree and homogeneous. Sa3+4ab+c is not homogeneous.

22. A vinculum or bar, or a parenthesis (), is used to express that all the terms of a polynomial are to be considered together. Thus,

a+b+cxb, or (a+b+c) xb,

denotes that the trinomial a+b+c is to be multiplied by b; also, a+b+cxc+d+f, or (a+b+c)x(c+d+f), denotes that the trinomial a+b+c is to be multiplied by the trinomial c+d+f.

When the parenthesis is used, the sign of multiplication is usually omitted. Thus,

(a+b+c) xb is the same as (a+b+c)b.

QUEST.-20. What is the dimension of a term? What is the degree of a term? How many factors in 3abc? Which are they? What is its degree? 21. When is a polynomial homogeneous? Is the polynomial 2a3b+3a2b2 homogeneous? Is 2a4b-b3 22. For what is the vinculum or bar used? Can you express the same with the parenthesis?

23. The terms of a polynomial which are composed of the same letters, the same letters in each being affected with like exponents, are called similar terms.

Thus, in the polynomial.

7ab+3ab-4a3b2+5a3b3,

the terms 7ab, and 3ab, are similar: and so also are the terms -4a3b2 and 5a3b2, the letters and exponents in both being the same. But in the binomial 8a2b+7ab2, the terms are not similar; for, although they are composed of the same letters, yet the same letters are not affected with like exponents.

24. When an algebraic expression contains similar terms, it may be reduced to a simpler form.

1. Take the expression 3ab+2ab, which is evidently equal to 5ab.

2. Reduce the expression 3ac+9ac+2ac to its simplest form. Ans. 14ac.

3. Reduce the expression abc+4abc+5abc to its simplest form.

In adding similar terms together we take the sum of the coefficients and annex the literal part. The first term, abc, has a coefficient 1 understood, (Art. 12).

abc

4abc

5abc

10abc

25. Of the different terms which compose a polynomial, some are preceded by the sign +, and the others by the sign The first are called additive terms, the others, subtractive terms.

QUEST.-23. What are similar terms of a polynomial? Are 3a2b2 and 6a262 similar? Are 2a2b2 and 2a3b3? 24. If the terms are positive and similar, may they be reduced to a simpler form? In what way?

The first term of a polynomial is commonly not preceded by any sign, but then it is understood to be affected with the sign +.

1. John has 20 apples and gives 5 to William : how many has he left?

Now, let us represent the number of apples which John has by a, and the number given away by b: the number he would have left would then be represented by a a-b.

2. A merchant goes into trade with a certain sum of money, say a dollars; at the end of a certain time he has gained dollars: how much will he then have ?

Ans. a+b dollars. If instead of gaining he had lost b dollars, how much would he have had? Ans. a-b dollars. Now, if the losses exceed the amount with which he began business, that is, if b were greater than a, we must prefix the minus sign to the remainder to show that the quantity to be subtracted was the greatest.

Thus, if he commenced business with $2000, and lost $3000, the true difference would be -1000: that is, the subtractive quantity exceeds the additive by $1000.

3. Let a merchant call the debts due him additive, and the debts he owes subtractive. Now, if he has due him $600 from one man, $800 from another, $300 from another, and owes $500 to one, $200 to a second, and $50 to a third, how will the account stand? Ans. $950 due him.

4. Reduce to its simplest form the expression

3a2b+5a2b—3a2b+4a2b—6a2b—a2b,

-.

QUEST.-25. What are the terms called which are preceded by the sign+? What are the terms called which are preceded by the sign — If no sign is prefixed to a term, what sign is understood? If some of the terms are additive and some subtractive, may they be reduced if similar? Give the rule for reducing them. Does the reduction affect the exponents, or only the coefficients?

Hence, for the reduction of the similar terms of a polynomial we have the following

RULE.

I. Form a single additive term of all the terms preceded by the sign plus; this is done by adding together the coefficients of those terms, and annexing to their sum the literal part.

II. Form, in the same manner, a single subtractive term. III. Subtract the less sum from the greater, and prefix to the result the sign of the greater.

REMARK.-It should be observed that the reduction affects only coefficients, and not the exponents.

EXAMPLES.

1. Reduce to its simplest form the polynomial

+2a3bc2-4a3bc2+6a3bc2—8a3bc2+11a3bc2.

Find the sum of the additive and subtractive terms separately, and take their difference: thus,

Hence, the given polynomial reduces to

19a3bc2-12a3bc2=7a2bc2.