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Difference of two Powers divisible by Difference of their Roots.

49. Theorem. The difference of two integral positive powers of the same degree is divisible by the difference of their roots.

an b" is divisible by a- - b.

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Thus,

Demonstration. Divide an

bn by a-b, as in art. 42,

proceeding only to the first remainder, as follows.

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1st Remainder = an−1 b —b" = b (an−1 — bn−1).

Now, if the factor a"-1 -bn-1 of this remainder is divisible by ab, the remainder itself is divisible by a—b, and therefore an -b" is also divisible by a -b; that is, if the proposition is true for any power, as the (n - 1)st, it also holds for the nth, or the next greater.

But from examples, 10, 11, 12, 13 of art. 43, the proposition holds for the 2d, 3d, 4th, and 5th; and therefore it must be true for the 6th, 7th, 8th, &c. powers; that is, for any positive integral power.

50. Corollary. The division of an

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by a

-b may be continued for the purpose of showing the form of the quotient,

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an · an−1b| an—1+an-2b+an-3 b2+ &c..... +ab"-2+bn−1

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Division of Polynomials.

that is,

an

bn

a b

=an−1+an-2b+an-3 b2+&c..... +abn−2+bn−1,

so that each term of the quotient is obtained from the preceding term by diminishing the exponent of a by unity and increasing that of b by unity; and the number of terms is equal to the exponent n.

51. Corollary. If b is put equal to a in the preceding quotient, each of its terms becomes equal to an−1, which gives the peculiar result

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52. There are sometimes two or more terms in the divisor, or in the dividend, or in both, which contain the same highest power of the letter according to which the terms are arranged.

In this case, these terms are to be united in one by taking out their common factor; and the compound terms thus formed are to be used as simple ones. It is more convenient to arrange the terms which contain the same power of the letter in a column under each other, the vertical bar being used as in art. 17; and to arrange the terms in the vertical columns according to the powers of some letter common to them.

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1. Divide a2 23-b2 x3 — 4 a b x2-2 a2x+2a bx +

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x3.

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a

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0.

In this quotient, the coefficient a+b of 22, the coefficient -b of x and the term a+b are successively obtained. by dividing the coefficient a2-62 of 23 in the dividend, the coefficient a2 - 2 a b + b2 of x2 in the first remainder, and the coefficient a2+2ab-b2 of x in the second remainder, by the coefficient a-b of x in the divisor.

--

Ans. (a+b)x+(a—b) x — (a—b).

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2. Divide (66 10) a1 — (7 b2 —23b+20) a3 (3 63 -226231b-5) a2+(4 b3—9 b25b-5) a+b2 -2b by (3b-5) a + b2 —2 b.

Ans. 2 a3-(3b-4) a2+(4 b-1) a + 1.

3. Divide a (b2 — 2 c2) a1 + (b1 — c1) a2 + (b® + 2b4c2b2c4) by a2-b2- c2.

Ans. — a1 — (2 b2 — c2) a2 — b1 — b2 c2.

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Terms of a fraction may be multiplied or divided by the same quantity.

CHAPTER II.

FRACTIONS AND PROPORTIONS.

SECTION I.

Reduction of Fractions.

54. When a quotient is expressed by placing the dividend over the divisor with a line between them, it is called a fraction; its dividend is called the numerator of the fraction, and its divisor the denominator of the fraction; and the numerator and denominator of a fraction are called the terms of the fraction.

When a quotient is expressed by the sign (:) it is called a ratio; its dividend is called the antecedent of the ratio, and its divisor the consequent of the ratio; and the antecedent and consequent of a ratio are called the terms of the ratio.

55. Theorem. The value of a fraction, or of a ratio, is not changed by multiplying or dividing both its terms by the same quantity.

Proof. For dividing both these terms by a quantity is the same as striking out a factor common to the two terms of a quotient, which, as is evident from art. 35, does not affect the value of the quotient. Also multiplying both terms by a quantity is only the reverse of the preceding process, and cannot therefore change the value of the fraction or ratio.

56. The terms of a fraction can often be simplified

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