Quantities in juxtaposition, without any sign between them, are to be multiplied together. Thus ab denotes a and b multiplied toge ther; and axy denotes a, x, and y multiplied together.

(10.) The sign by between two quantities, denotes that the quantity before the sign is to be divided by the one after it. Thus a÷b, a by b, denotes that a is to be divided by b.

Division is also denoted by placing the dividend over the divisor, with a line between them, after the manner of a Fraction;

An integral quantity, in Algebra, is one which does not express any operation in division, whatever may be the numerical values which the letters represent.

(11.) A parenthesis ( ) enclosing an algebraic expression, or a vinculum drawn over it, connects the value of that expression with the sign which immediately precedes or follows it.

Thus (a+b).c, or (a+b)c, a plus b in a parenthesis into c, denotes that the sum of a and b is to be multiplied into c.

The same thing would be denoted by a+bxc, a plus b under a vinculum into c.—In a+bc, only b would be multiplied into c.

The vinculum, and the expression affected by it, are sometimes set vertically.

In the elementary oral Exercises which are occasionally inserted, the Student should write down the quantities as they are read to him. Suppose the letters a, b, c to represent the numbers 3 4 5, respectively; then what is the numerical value of a+b-c?

What is the value of ab+c?
What is the value of ac÷b?
What is the value of bc÷a?

Of abc-bc+a?

Of ab+b-ac?

Of abc+ac-bc?

What is the value of (a+b) c, a plus b in a parenthesis into c?

What is the value of (ab+c) a?
What is the value of (a+b)÷c?
What is the value of (bc-a)÷b?

The Teacher may propose other Exercises he deem it necessary.

Of (a+b+c) c?
Of (ab+b-c) a?
Of (bc-a+b)b?

of the same nature, should

Factors.-Constant Product.

(12.) Two or more quantities multiplied together, are called the factors of their product; and the Product is the same in value, in whatever order its factors are taken.

Thus a and x are the factors of the product ax; and this product is the same in value as xa. So a, b, and c are the factors of the product abc, or acb, or bca, &c.

It is most convenient to set literal factors according to the order of the same letters in the Alphabet; thus ax; abc.

To understand why the product ax is equal to xa, consider a and x as representing numbers, and that the product of two numbers is the same, when either of them is made the multiplier.

For example, 25 times 7 is equal to 7 times 25. For 25 times 7 must be 7 times as many as 25 times 1, which is 25; that is, 25 times 7 is equal to 7 times 25.

Prove that 14 times 9 is equal to 9 times 14.
Prove that 31 times 11 is equal to 11 times 31.

Prove that 23 times 15 is equal to 15 times 23.

Prove that 47 times 18 is equal to 18 times 47.

Powers and Roots.

(13.) The first power of a quantity is the quantity itself; thus the first power of 5 is 5, and the first power of a is a.

The second power, or square, of a quantity, is the product of the quantity multiplied into itself. Thus the second power, or square, of 5 is. 5 x5, which is 25; and the second power of a is aa.

The third power, or cube, of a quantity is the product of the quantity multiplied into its second power, or square. Thus the third power, or cube of 5 is 5×5×5, which is 125; and the third power of a is aaa.

What is meant by the fourth power of a quantity? What is meant by the fifth power of a quautity? By the seventh power of a quantity?

What is the square of 3?

The cube of 4? The fourth power of 2? The square of 7? The cube of 6? The fourth power of 10?

(14.) The second root, or square root, of a quantity, is that quan tity whose square is equal to the given quantity. Thus the square root of 9 is 3; and the square root of aa is a.

The third root, or cube root, of a quantity, is that quantity whose third power, or cube, is equal to the given quantity. Thus the cube root of 8 is 2; and the cube root of aaa is a.

What is meant by the fourth root of a quantity? What is meant by the fifth root of a quantity? By the ninth root of a quantity?

What is the square root of 16? root of 16? The square root of 81?

The cube root of 27? The fourth
The cube root of 1000?

The cube of the cube

What is the square of the square root of 4? root of 125? The square of the cube root of 64? square root of 16?

The cube of the

Coefficients and Exponents.

(15.) The coefficient of a quantity is any multiplier prefixed to that quantity. In a more general sense, the coefficient of a quantity is any factor forming a product with that quantity.

Thus, in 3a, 3 is the coefficient of a, and denotes 3 times a. In 5ax, 5 is the coefficient, denoting 5 times ax. In x, is the coefficient of x, and denotes one-half of x.

When no numerical coefficient is prefixed, a unit is always to be understood. Thus a is 1a, once a, and ax is lax, once ax.

(16.) The exponent of a quantity is an integer annexed to it, to denote a power, or a fraction annexed to denote a root, of that quantity. Thus a2, a with exponent 2, denotes the second power, or square of a; x3 denotes the third power, or cube, of x; and so on.

The fractional exponents,,, and so on, denote, respectively the square root, cube root, &c., of the quantity to which they are annexed.

a, a with exponent, denotes the square root of a; the cube root of x; and so on.

2

denotes

When no exponent is annexed to a quantity, a unit is always to be understood. Thus a is a1, the first power of a. (13.)

An exponent is assigned to the product of two or more factors, by affecting such product with a parenthesis, or a vinculum, and the exponent.

Thus (ax)2 or ax2, ax in a parenthesis, or under a vinculum, with exponent 2, denotes the square of the product ax; whereas ax2 denotes a into the square of x.

(17.) An integral coefficient indicates the repeated addition of a quantity to itself; while an integral exponent indicates the repeated multiplication of a quantity into itself.

Thus 3a, 3 times a, is equivalent to a+a+a;. while

a3, the third power of a, is equivalent to aaa.

Coefficients and exponents are thus employed to abbreviate the language of Algebra.

Zabc?

What is the equivalent, in Addition, of 2x? Of 3ax? Of Of 5a2? Of 4ay?? Of 3axy? What is the equivalent, in Multiplication, of a?? Of x3? Of a2? ax2? Of a2x? Of abx3? Of ac2x2?

Allowing the value of a to be 4, what is the value of a2? Of a?

Of a3? Of 5a2? Of 10a? Of a1a3? Of (4a)* ?

Allowing a to be 4, and 6 9, what is the value of ab? Of ab?

1

Of a2b1? Of (ab)? Of ab? Of Jab? Of {(ab)*?

Similar and Dissimilar Quantities.

(18.) Similar quantities are such as have all the literal factors, with their respective exponents, the same in each; otherwise, the quantities are dissimilar.

Thus the two quantities 2ax2 and 5ax2 are similar; while 3ax2 and 4a2x are dissimilar, the literal factors not having the same exponents in each.

Are the two quantities al and 3ab similar, or dissimilar? Are 4a and 3x similar, or dissimilar? Are ay and 3ya similar, or dissimilar? 5ax and 7ax? abc2 and 5bc2? 2ab and 2ab2? 3axy2

and ay2x? 2bc3 and 36c2?

Give an example of three similar quantities.-Give an example of three dissimilar quantities.-Another example of three similar quantities.--Another example of three dissimilar quantities.

Monomials and Polynomials.

(19.) An algebraic monomial is a symbol of quantity not composed of parts connected by the sign + or —.

Thus 3a, 5ax, 2x2, and abc3 are monomials.

(20.) An algebraic polynomial consists of two or more monomials connected by the sign + or -; and such monomials are called the terms of the polynomial.

Thus 5a2+bx, and ax2 +36-5c2 are polynomials.

A polynomial composed of two terms is called, more definitely, a binomial, and one composed of three terms, a trinomial.

(21.) The value of a polynomial is not affected by changing the order of its terms, without changing the sign prefixed to any term.

Thus a+b-c is equivalent to a-c+b; for the result will evidently be the same, whether b be first added to a, and c then subtracted; or c be first subtracted, and b afterwards added.

(22.) A polynomial is arranged according to the powers of one of its letters, when the exponents of that letter increase, or decrease, continually in the successive terms.

Thus the polynomial 3a3+4a2x-5ab, is arranged according to the descending powers of a, since the exponents of a decrease continually in the successive terms.

The letter, as a in this example,―according to which the polynomial is arranged, is called the letter of arrangement.

(23.) A polynomial is said to be homogeneous, when the sum of the exponents of the literal factors, is the same in each of its terms. Thus a3+2ax2-bcy is homogeneous, since the sum of the exponents of the literal factors is the same, namely 3, in each term.

Each one of the literal factors composing a term, is called a dimension of that term; and the degree of any term is the ordinal of the number of its literal factors or dimensions.

Thus 4a2x contains three literal factors, aax, and is therefore of three dimensions, or of the third degree.

Arrange the polynomial 2x+3a2x2-4ax3+a3y2. according to the descending powers of a.-Arrange it according to the ascending powers of a.—Which of the terms of this polynomial are homogeneous? Tell the number of dimensions in each term.-Of what degree is each term?