9. Divide a3 + 5a3x + 5ax2 + x3 by a + x.

ALGEBRAIC FRACTIONS have the same names and rules of operation, as numeral fractions in common arithmetic; as appears in the following Rules and Cases.

CASE I.

To reduce a Mixed Quantity to an Improper Fraction.

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.

To reduce an Improper Fraction to a Whole or Mixed

Quantity.

DIVIDE the numerator by the denominator, for the integral part; and set the remainder, if any, over the denominator, for the fractional part; the two joined together will be the mixed quantity required.

3

b

to mixed quantities.

First, 16 = 16 ÷ 3 = 51, the answer required.

a2

a3

And, ab + = (ab + a2) ÷ b = a + — — • Answer.

2ac 3a2 3ax +4x2

and

a + x

(2ac3a2)+c=2a- Answer.

2. To reduce

to mixed quanti

2ac 3a2

3a2

First,

=

C

MULTIPLY every numerator, separately, by all the denominators except its own, for the new numerators; and all the denominators together, for the common denominator.

When the denominators have a common divisor, it will be better, instead of multiplying by the whole denominators, to multiply only by those parts which arise from dividing by the common divisor. Observing also the several rules and directions, as in Fractions in the Arithmetic.

bx

a b az

Here and = and

, by multiplying the terms of

the first fraction by z, and the terms of the 2d by x.

2. Reduce

α x

Here

b abc cx2

b2x

and = and by multiplying the x' b' C bcx' bcx' bcx'

terms of the 1st fraction by bc, of the 2d by cx, and of the 3d by bx.

5a 36

5. Reduce and and 4d, to a common denominator.

3x

2c'

5

6

3a

4

3a

6. Reduce and and 26+ to fractions having a com

b'

To find the greatest common Measure of the Terms of a Fraction.

DIVIDE the greater term by the less, and the last divisor by the last remainder, and so on till nothing remains; then the divisor last used will be the common measure required; just the same as in common numbers.

But note, that it is proper to range the quantities according to the dimensions of some letters, as is shown in division. Note also, that all the letters or figures which are common to each term of the divisors, must be thrown out of them, or must divide them, before they are used in the operation.

Therefore the greatest common measure is a + b.

2. To find the greatest common measure of

a3-ab2

a2+2ab+b2

Therefore a+b is the greatest common divisor.

a2-4

3. To find the greatest common divisor of

4. To find the greatest common divisor of

Ans. a2-b3.

5. Find the greatest com. measure of

a'r+2a2x2+2ax3+x1 5a+10a x+5a3x2*

CASE V.

To reduce a Fraction to its lowest Terms.

FIND the greatest common measure, as in the last problem. Then divide both the terms of the fraction by the common measure thus found, and it will reduce it to its lowest terms at once, as was required. Or divide the terms by any quantity which it may appear will divide them both as in arithmetical fractions.

Here ab+b2 is divided by the common factor b.

Therefore a + b is the greatest common measure, and

ab+b2

b

hence a+b) ac2+bc2= c'

is the fraction required.

c2+2bc+b3

to its least terms.