(ART. 30.) When the algebraic expression is a polynomial, and has prime factors that are monomials, such monomial factors are visible, as in the following expressions:

Thus in the first expression, x is visible in every term, it is, therefore, a common factor to every term, and (1+a) is the other factor, and the product of these two factors makes the expression; and so for the other expressions.

The examples in division (Art. 23), are analagous to these, except that in that article the divisor is given, and may not be contained in every term, as in example 7, (Art. 23).

(ART. 31.) When all the prime factors composing any algebraic expression consist of binomials or polynomials, they are not visible in the expression like a monomial, and we can find them only from our general knowledge of algebraic expressions.

For instance, the prime factors in the expression (a2+2ab +62) we know to be (a+b) and (a+b) by (Art. 16), and all other expressions that correspond to a binomial squared, is immediately recognized after a little experience in algebraic operations.

Also, any expression which is the difference of two squares, as (a2-62) is instantly recognized as the product of the two prime factors, (a+b), and (a—b), (Art. 17).

The expression ax+ay+bx+by can be resolved into two prime factors, by inspection, thus, a(x+y)+(x+y) is merely a change in the form of the expression. Now put (x+y)=S. Then the next change is aS+bS; the next is (a+b)S. Restoring the value of S, we have (a+b)(x+y) for the prime factors in the original expression.

(ART. 32.) Any trinomial expression in the form of ax2+ bx+c, can be resolved into two binomial factors; but the art of finding the factors is neither more nor less than resolving an equation of the second degree, a subject of great importance and some difficulty, which will be examined very closely in a subsequent part of this work; therefore it is improper to treat upon this subject at present.

(ART. 33.) Common multiple, and least common multiple, have the same signification in Algebra as in Arithmetic, and are found by the same rule, except changing the words number and numbers in the rule for quantity and quantities.

Or, we may take the following rule to find the least common multiple in algebraic quantities.

RULE -1. Resolve the numbers into their prime factors.

2. Select all the different factors which occur, observing, when the same factor has different powers, to take the highest power. 3. Multiply together the factors thus selected, and their product will be the least common multiple.

EXAMPLES.

1. Find the least common multiple of 8a2x2y, and 12a3b3x. Resolving them into their prime factors,

8a2x2y=2°Xa2X x2 Xy

12a3b3x=22 a3 × × × b3 × 3

The different factors are 23, a3, x2, y, b3, 3, and their product is 24a3b3x2y, which is the least common multiple required. 2. Required the least common multiple of 27a, 15b, 9ab, and 3a2. Ans. 135a2b.

3. Find the least common multiple of (a2-a2), 4(a-x), (a+x). Ans. 4(a2x2). 4. Required the least common multiple of a2(a-x), and ax1 (a2—x2). Ans. a2x2(a2—x2).

5. Required the least common multiple of x2(x-y), a2x2, and 12axy2. Ans. 12axy(x—y). 6. Required the least common multiple of 10a2x2 (a—b), 152(a+b), and 12(-b). Ans. 60a2a (a—b2).

ALGEBRAIC FRACTIONS.

THE nature of fractions is the same, whether in Arithmetic or Algebra, and of course those who understand fractions in Arithmetic, can have no difficulty with the same subject in Algebra.

(ART. 33.) A fraction is one quantity divided by another when the division is indicated and not actually performed.

Hence every fraction consists of two parts, the dividend and divisor, which take the name of numerator and denominator. The numerator is written above a line, and the denominator

a

below it, thus,, and is read, a divided by b.

For illustration, we may consider any simple fraction as 3; here we consider one or unity divided into 5 parts, and 3 of these parts are taken. The 5 denotes the parts that the unit is divided into, hence it is properly named denominator, and the 3, numbers the parts taken, and is, therefore, properly called the numerator. So in the fraction, b denotes the

α

parts into which unity is divided, and a shows the number of parts taken.

In a numeral fraction, as, it is evident that if we double both numerator and denominator, we do not change the value of the fraction; thus, is the same part of the whole unit as

, and thus it would be if we multiplied by any other number; and conversely, we may divide both numerator and denominator by the same number, without changing the value of the fraction. Hence, if any fraction contains any factors common to both numerator and denominator, we may suppress them by division, and thus reduce the terms of the fractions to smaller quantities.

Hence, to reduce fractions to lower terms when possible, we have the following rule:

RULE.-Divide both terms by their greatest common divisor. Or, resolve the numerator and denominator into their prime factors, and then cancel those factors common to both terms.

Here 7abc is the common divisor, and dividing according

(ART. 34.) Fractions in Algebra, as in Arithmetic, may be simple or complex, proper or improper, and the same definitions to these terms should be given, as well as the same rules of operation; for in fact this part of Algebra is but a generalization of Arithmetic, and in some cases we give arithmetical and algebraical examples side by side.

A mixed quantity in Algebra is an integer quantity and a fraction; and to reduce these to improper fractions, we have the following rule:

RULE.-Multiply the integer by the denominator of the fraction, and to the product add the numerator, or connect it with its proper sign + or -; then the denominator being set under this sum, will give the improper fraction required.

EXAMPLES.

1. Reduce 23 and a+2 to improper fractions.

These two operations, and the principle that governs them,

are exactly alike.

2. Reduce 57 and a+a2 to improper fractions.

b