5.

tion, having reduced it to its lowest terms, or rejected the
leuters and factors, that are found in every term of the divi-
dend and divisor.

Thus, a3bx+acx2+ax3, divided by adx+anx,
abx-Lacx Laxs ab+cx+x2

gives

6. And a+ab+d, divided by a3-ac+a3c3, gives a+ab+ds

a3—ac+a3c3

Here the quotient cannot be reduced to lower terms, because the factor a is not to be found in the term da.

But it is to be observed, that though a fraction cannot be reduced to lower terms by a simple divisor, yet it may sometimes be so reduced by a compound one; as will appear the reduction of fractions.

in

7. Divide a3+x3 by a+x.

Ans. a3-ax+x3.

8. Divide a3-3α3y+зay3—y3by a-y.

Ans. a2ay+y3.

9. Divide 6x-96 by 3x-6.

Ans, 2x3+4x2+8x+16.

10. Divide a2-2ax+x2.

་་་་་

a—5a*x+10a3x2—10a3x3 +5ax*—x by

Ans. a3a3x+3ax3-x3.

FRACTIONS.

ALGEBRAIC FRACTIONS have the same names and rules of operation, as fractions in Arithmetic.

PROBLEM I.

To find the greatest common measure of the terms of a frac

tion.

RULE.

1. Range the quantities according to the dimensions of some letter, as is shown in division.

2. Divide the greater term by the less, and the last divisor by the last remainder, and so on till nothing remain ; then the divisor last used will be the common measure required.

NOTE. All the letters or figures, which are common to each term of any divisor, must be rejected before such divisor is used in the operation.

EXAMPLES.

1. To find the greatest common measure of

cx+x®)ca2+a3x

or c+x)ca3+u3x(a2

cas+a2x

cx+x2
ca2+a3x

Therefore the greatest common measure is c+x.

2. To find the greatest common measure of

x2+2bx+62

x2+20x+b2)x3-b2x(x

x2+2bx2 +b2x

1. Find the greatest common measure, as in the last prob

lem.

2. Divide both the terms of the fraction by the common measure thus found, and it will be reduced to its lowest

terms.

Therefore c+ is the greatest common measure ; and

cx+x3

is the fraction required.

Therefore x+b is the greatest common measure, and

To reduce a mixed quantity to an improper fraction.

RULE.

Multiply the integer by the denominator of the fraction, and to the product add the numerator; then the denomina. tor being placed under this sum will give the improper fraction required..

To reduce an improper fraction to a whole or mixed quantity.

RULE.

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