2. To find the product of a2 by a3.

a2 · a3 =

(a・ a) (a・ a⋅ a) a⋅ a a· a⋅ a = a3.

The exponent 5 is the sum of the exponents 2 and 3.

3. To find the product of am by a" where m and n are positive

aTM · a" — (a · a · a tom factors). (a a a ton factors)

= a a· a· • to (m + n) factors (by IV, 7)
= am+n (by 88).

The final exponent ("+") is the sum of the exponents m and " of the factors of the product.

Hence, the exponent of a letter in a product is equal to its exponent in the multiplicand plus its exponent in the multiplier.

10. The Index Law for multiplication may be extended to the product of several powers of a number a, thus,

By means of the laws of indices and of multiplication we can simplify products as follows:

13 ab2ca3c27b7

13 a· a3b2bc1c2 = 91 a13c1+2;

8 amcndra'cp = 8 amalc"cPdr 8al+mcn+pd";

=

In the product 12 abcd, 12 is called a numerical factor and a3, b, c, d, and abcd2 are called literal factors.

DEFINITIONS

11. A product may consist of a numerical factor and a literal factor; in this case the number represented by the numerical factor is called the coefficient of the latter. Thus in the product 7 abc the factor 7 is called the coefficient of the factor abc. Where there is no numerical factor, we may supply unity; thus we may say that, in the product abc, the coefficient is unity.

In case the product consists entirely of literal factors, any one factor may be called the coefficient of the product of the remaining factors. Thus, in the product abc, we may call a the coefficient of

bc, or b the coefficient of ac, or c the coefficient of ab. If it is necessary to distinguish these two kinds of coefficients, the latter may be called literal coefficients, and the former numerical coefficients.

Of the results at the end of 10 it may be said:

12, 91, 8, and 2 are respectively the numerical coefficients of a3b*c2d2, a1b3c1+2, al+mcn+pd", and Aa13c.

12. Monomial is the name given to a single factor or the product of two or more factors; for example,

5 a, 7abe, 9 a3b3c2, amb"+c", etc.

If two or more monomials be connected by one or more of the operations of Algebra--multiplication, etc.-the result is called an algebraic expression; thus:

5a7abc, 6a2+(9 a3b2c+ambn+oc”), etc.

The monomials 5a, 7 abc, 6a2, 9 a3b2c, amb+Pcr are called the terms of the algebraic expressions,

5 a 7 abc and 6 a2+(9 abc + ambn+Pc").

Positive Terms are those which have the plus sign prefixed, e. g., +7 abc,+amb"+Pe". If no sign precedes a term the plus sign is understood; thus 5 a, 6 a2, 9 a3b2c are respectively the same as +5 a, +6a2, +9 a3b3c.

13. Similar or Like Terms are those which do not differ at all, or differ only in their numerical coefficients; otherwise terms are said to be unlike. Thus 3a, 5 ab, 7 a2, and 6 abe are respectively similar to 15 a, 9 ab, 11 a2, and 13 a2bc. And ab, a2b, ab2, and abc are all unlike.

14. Each letter which occurs in an algebraic product is called a dimension of the product, and the number of the letters is the degree of the product. Thus ab3c2 or a· b⋅ b b c c is said to be of six dimensions or of the sixth degree. A numerical coefficient is not counted; thus 9 a3b1 and a3b1 are of the same dimensions, namely seven dimensions. Hence the degree of a term or the number of dimensions of a term is the sum of the exponents. It should be remembered that if no exponent is expressed the exponent 1 is understood as indicated in 8.

15. An algebraic expression is said to be homogeneous when all its terms are of the same degree or dimensions. Thus 5a3a2b3+

9a be2 is homogeneous, for each term is of five dimensions.

16. Addition of Similar Monomials.—(13.)

1. The sum of 7 a and 9 a is required.

Hence, to add two positive similar terms, find the sum of their coefficients (11) and affix to the result the common letters.

2. Find the sum of 9 abc and 16 ab3c.

By the rule above,

9 abic +16 abc = (9 +16) ab3c

= 25 ab°c.

17. Addition of Polynomials of Plus Terms.-A polynomial is an algebraic expression of two or more terms.

The addition of polynomials is accomplished by means of the second law of addition (II, 226 and 16, Rule).

EXAMPLE. Find the sum of

6 a +9 x2, 3x2+5 a + 6 y3, and 2x2+a +mn.

It is convenient in practice to write the expressions one underneath the other, with similar terms arranged in the same column. Find the sum of the terms in each column (816), and write the results connected with the plus sign.

(by index, commutative, and associative laws of multiplication.) Similarly, the product of 9 ab'cmd" by 13 ab3crd" is

=

9 ab'cmd" × 13 a1b3cod” = 9 · 13 · a·abb3ç3ç3d"d" — 117 ak+1fl+3cm+Pdn+r. The coefficient 55 of the resulting product 55abc is the product of the two coefficients 5 and 11 of the multiplicand 5 a3c and the multiplier 11ab3c; the literal part ab3 is the result of forming a product of all the different letters occurring in both multiplicand and multiplier, each with an exponent equal to the sum of the exponents of their letters in both multiplicand and multiplier. The product

117 ak+1b1+3cm+Pd+r is formed in a similar manner. The coefficient 117 is the product of the two coefficients 9 and 13, and the exponents +1 1+3 m+p

,

m+p, n+r are respectively the sums of the exponents of From 27, laws III, IV for multiplication, and 18, the following rule for the product of two monomials is derived:

a, b, c, d in both the multiplicand and multiplier.

To the product of the two monomial coefficients (11) annex the letters, each with an exponent equal to its exponent in the multiplicand plus its exponent in the multiplier.

EXAMPLE. Multiply 7 xyz by 3x3y5; m being a positive integer. By rule 7 xTMyz× 3x3y3=7 · 3xm+3y1+5x= 21 xm+3y3z.

19. The Multiplication of a Polynomial of Plus Terms by a Plus Monomial. From 27, Law V,

Multiply each term of the multiplicand by the multiplier, and add the partial products.

EXAMPLE.-Multiply 2x3+5x+7 by 7 x2.

By the rule above,

(2x3+5x+7) × (7 x2) = (2 x3 + 5x)7 x2+7(7 x2)

= (2.3) (7.*)+(5) (7)+7(7)

= 14x+35 x3+ 49 x2.

The following exercises will serve as illustrations of the preceding definitions and rules of addition and multiplication.

EXERCISE I

If a = 1, b = 3, c = 4, d= 6, e = 2, ƒ 0, x = 3, y=3, find the numerical values of the eight following algebraic expressions:

and

1. Find the value of a +2b+4 c; here, a = 1, b = 3, c=4,

2. ab+2 bc + 3ed.

a+2b+4c=1+2 · 3+ 4·4 = 1 + 6 + 16 = 23.

3. ac+4cd+3 cb.

Add the following (see 16, 17):

9. 2x+5x+x+7, 3x2+2+6x3+8x, x+3x+4, and 1+2x2 + 5.x. 10. 2a+3b4d, 2b+3d+4c, 2d+3c+4a+4b, and 2c+3a. 11. a2+3xy + y2+ x + y + 1, 2 x2+4 xy +3y2+2x+2y+3, 3x2+5xy+ 4y2+3x+4y+2, and 6x2+10xy+5 y2+x+y.

12. 2x3+ax2, x3 +3 ax2, x3+2ax2+ a2x.

13. 4x+10a3+ux(5x+6a), 3(2a3+x3)+2ax(2x+a), x2(17x+19 a) + 15 ar, and 62 (x+3a)+a2 (7x+5a).

(Remove parentheses by Law V, ¿7.)

14. 4 ab+x2, 3x2+2 ab, 2x (a+b) and 5 a (x +7b)+11x(x+13b+7a).

Find the following products by ??18, 19:

15. 3axy and 5ay1.

16. 2a, 3b, 4c, ab, bc, and abc.

17. Simplify (3 xy2) · (5.xyz3) · (4 y2 zw) · ( x y z w).

18. Find the product of:

x+y+2 and 2xyz

3x2+5y3+725 and 9 xy223

3x+2a (x+2ay+522) and 5 a2xy2z.