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113. And if any two unequal powers of the same root be taken, it is plain, from what is here shown, that it -x", or "(**--— 1)
...(7), is divisible by x-1,
whether m -n be even or odd; and that
am—x, or x"(**--— 1)......(8), is divisible by <+1, when m-n is an even number; as also that
am+x", or x"(xm--*+1)......(9), is divisible by x+1, when man is an odd number.
114. It is very proper to remark, that the number of all the factors, both equal and unequal, which enter in the formation of any product whatever, is called the degree of that product: The product a2b3c, for example, which comprehends six simple factors, is of the sixth degree; this, a?hac is of the tenth degree; and so oil.
Also, that if all the terms of a polynomial, or compound quantity, be of the same degree, it is said to be homogeneous. And, it is evident from the rules established in Multiplication, that
if two polynomials be homogeneous ; their product will be also homogeneous ; and of the degree marked by the sum of the numbers which designate the degree of those factors.
Thus, in Ex. 1, page 39, the multiplicand is of the fourth degree, the multiplier of the third, and the product of the degree 4+3, or of the seventh degree.
In Ex. 12, page 42, the multiplicand is of the third degree, the multiplier of the third, and the product of the degree 3+3, or of the sixth degree.
Hence, we can readily discover, by inspection only, the errors of a product, which might be committed by forgetting some one of the factors in the partial multiplications.
115. We have seen in the division of two simple quantities (Art. 84), that when certain letters, factors in the divisor, are not common to the dividend, and reciprocally, the division can only be indicated, and then the quotient is represented by a fraction whose numerator is the product of all the letters of the dividend, not common to the divisor, and denominator, all those letters of the divisor, not common to the dividend. Let, for example, abmn be divided by cdmn; then,
may whole number for certain numeral values of the letters a, b, c, and d; thus, if we had a=4, b=6, c=2, d=3; but that, generally speaking, it will be à numerical fraction which can be reduced to a more simple expression.
§ 1. Theory of Algebraic Fractions.
116. It is evident.(Art. 103), that if we perform
the same operation on each of the two members of an equality, that is, upon two equivalent quantities or numbers, the results shall always be equal.
It is by passing thus from the fractional notation to the algorithm of equality, that the process to be pursued in the researches of properties and rules, becomes simple and uniform. 117. Let therefore the equality be
a=bXv...i (1), when we divide both sides by b which has no factor common with a, we shall bave
(2). b Thus v will represent the value of the fraction b
or the quotient of the division of a by b. 118. If the numerator and denominator of a fraction
be both multiplied, or both divided by the same quantity, its value will not be allered.
For, if we multiply by m the two members of the equality (1), we will have these equivalent results, ma=mb xo
(3); dividing both by mb, we shall have
(4), mb b m being any whole or fractional number whatever. 119. If a fraction is to be multiplied by m, it is the
same whether the numerator be multiplied by it, or the denominator divided by it.
For, if we divide by b, the two members of the equality (3), we obtain the following, =mXo
The equality (1) may also be put under the form
1 whence we derive, dividing each side by--b,
120. If a fraction is to be divided by m, it is the same
whether the numerator be divided by m, or the denominator multiplied by it. For, from the equality (1), we deduce these (3)
-=bx a=m6 x (9), dividing the first by b and the second by mb, in order to have
- they become
- (11). It is to be observed that in 7, the numerator is and the denominator b, and that we employ the greater line for separating the numerator from the denominator.
121. If two fractions have a common denominator,
their sum will be equal to the sum of their numerators divided by the common denominator. For, let now the two equalities be (12)....a=bXv; a'=bXv.... (13)
' corresponding to the fractions
which have the same denominator; adding the two equalities (12) and (13), we shall have
ata'=bo+bo'=(t+o'); and dividing both members by b, in order to have the sum sought otu, it becomes
= =oto'.... (14).
b Note. In adding the above equalities, the corresponding members are added; that is, the two members on the left-hand side of the sign =, are added together, and likewise those on the right. The same thing is to be understood when two equalities are subtracted, multiplied, &c. 122. If two fractions have a common denominator,
their difference is equal to the difference of their numerators divided by the common denominator.
For, if we subtract equality (13) from (12), we shall have
a-a-bo-bo'=6(0-0'); dividing each side by b, and we will obtain
(15), 6 123. Let us suppose that the two fractions have different denominators, or that we have the cqualities
a=b .v, a'=' .2'; we will multiply the two members of the first by b', and those of the second by b, an operation which will give
ab'=bb'v, a'b=bb'o'; then adding and subtracting, we have
ab'+a'b=bb'(v+v'), the double sign + which we read plus or minus, indicating at the same time both addition and subtrac