Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

Examples.

2. Divide a +5a*x+10a3r© +- 10a2x2+5ax*+x3.

Ans. a + 3 a® x + 3 ax2 + r.

[ocr errors]

3. Divide 12 x3- 13 x 34 x + 35 x by 4x-7x.

[blocks in formation]

— xo 5. Divide y 3y'x2 + 3y2x2 + x by y3-3yx + 3yx2 — 23.

[merged small][ocr errors][merged small][merged small]

Ans. y + 3yx + 3yx2 + x3 10a3x3 + 5ax1 — x3 by a' Ans, as ·3ax +3ax1

Miscellaneous Examples.

r.

1. Divide a3-r3 by a — x. Ans, a + a'r + a2r2 + ax3 + x*.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

6. Divide r3 + 3x3y + 3xy2 + y3 by x2 + 2xy + y2.

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Divide 6x96 by 3r - 6. Ans. 2x3 + 4x2 + 8x + 16.

8.

[merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

Algebraic Fractions are subject to the same rules as are the fractions of common arithmetic, and their reduction requires only the application of the principles laid down in the preceding rules.

PROBLEM I.

To reduce a mixed quantity to an improper fraction.

Rule. Multiply the integer by the denominator of the fraction, and to the product annex the numerator with its proper sign; under this same place the former denominator, and the result will give the improper fraction required.

[blocks in formation]

Here the integral part × denom. of the fraction + the numerato? = 3a x 5a2 + 2x = 15a3 + 2x.

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

To reduce an improper fraction to a whole or mixed quantity. Rule. Observe which terms of the numerator are divisible by the denominator without a remainder, the quotient will give the integral part; to this annex the remaining terms of the numerator, with the denominator under them, with their proper signs, and the result will be the mixed quantity required.

Example 1.

to a whole or mixed quantity.

Reduce

a2 + ab + b2

a

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small]

To reduce fractions to a common denominator.

Rule. Multiply each numerator into every denominator but its own for the new numerators, and multiply all the denominators for

a common denominator.

5x multiplied by 7 = 35x ; and 2x-3 subtracted from 35% leaves 33x+3

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

To reduce a fraction to its lowest terms.

Rule. Observe what quantity will divide all the terms of both numerator and denominator without any remainder: divide them by this quantity, and the fraction is reduced to its lowest terms.

1. Reduce

Examples.

14237ax + 2x2

35 x

to a common denominator.

Note. The co-efficient of every term of both numerator and denominator

is divisible by 7, and a enters also into every term; +7x will therefore divide both numerator and denominator without a remainder.

For 14x+7ax + 21.x2

[blocks in formation]

= 2x2 + a + 3x ;

=5x. Hence, the frac. in its lowest terms, is

2a2 + a +3

7x

5x

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

a2 + ab + b2

as

Scholium. When, therefore, we have occasion to reduce algebraic fractions to their lowest terms, by finding the greatest common measure of the numerator and the denominator, as in common arithmetic, it is evident that the rule which we have given for the solution of the foregoing problem answers every practi cal purpose, since it is only requisite to discover what quantity will divide both numerator and denominator, and that the quantity must therefore be the greatest common measure.

PROBLEM V.

(43.) To add fractional quantities.

Rule. Reduce the fractions to a common denominator, as in Problem III. Add all the numerators together, and below their sum set down the common denominator, and it will give the sum of the fractions required.

Examples.

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small]

(44.) To subtract fractional quantities.

Rule. Reduce the fractions to a common denominator as in Problem III; and then subtract the numerators from each other, and under the difference, write the common denominator.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]
« ΠροηγούμενηΣυνέχεια »