Ex. 6. Divide ax2+3a2x2-axy-3a'xy' by ax3-ay3. •

Ans. x+xy3+3ax.

Ans. 2a-3b+2c2.

Ex. 7. Divide 2a -5a'b+2a'c'+36-2bc' by a2-b.

Ex. 8. Divide a +2a'z'+z' by a2-az+z2.

Ans. a*+az+az3+z1.

Ex. 9. Divide ao-16a3x3+64x° by a2-4ax+4x2.

Ans.

Ex. 10. Divide a1+6a2x2-4a3x+x*-4ax3

by a1-2ax+x2.

Ans. a2-2ax+x3.

Ex. 11. Divide x*+x'y'+y' by x2+xy+y3.

Ans.

Ex. 12. Divide 12x-192 by 3x-6.

Ans. 4x+8x2+16x+32

Ex. 13. Divide a'+3a'b'-3a+b2-b°

by a'-3a2b+3ab2-b3.

Ans. a'+3a2b+3ab'+b2.

Ex. 14. Divide 2a'x+a'bx+acx-6a2b2-5abc-c"

by ax+2ab+c.

Ans. 2ax-3ab-c.

Ex. 15. Divide a +2ab'-2a'b-4b' by a-2b.

Ans. a3+2b3.

Ex. 16. Divide 4x-5x+12x-3 by 2x2+3x-1.

Ans. 2x2-3x+3.

Ex. 17. Divide 2y-19y'+26y-16 by 2y-3y+2.

Ans. y-8.

Ex. 18. Divide a°-2a'x3-2a'x+4x' by a'-2x.

Ex. 19. Divide a3-b' by a-b.

Ex. 20. Divide a-b' by a-b.

Ans. a*--2x3.

Ans.

Ans.

(77.) If the first term of the arranged dividend is not divisible by the first term of the arranged divisor, the complete division is impossible. So, also, the complete division is impossible when the first term of one of the remainders is not divisible by the first term of the divisor.

QUEST.-When will the complete division be impossible?

SECTION VI.

FRACTIONS.

(78.) WHEN a quotient is expressed as described in Art. 7, by placing the divisor under the dividend with a line between them, it is called a fraction; the dividend is called the numerator, and the divisor the denominator of the fraction. Algebraic fractions do not differ essentially from arithmetical fractions, and the same principles are applicable to both. The denominator shows into how many parts a unit is divided; and the numerator shows how many of those parts are used; or the denominator shows into how many parts the numerator is divided.

Thus, the fraction

6

11

indicates that a unit has been

divided into eleven equal parts, and that six of these parts are supposed to be taken.

a

So, also, the fraction indicates that a unit has

b

been divided into b equal parts, and that a parts are supposed to be taken; or the numerator a is to be divided into b parts.

Every quantity which is not expressed under a fractional form is called an entire quantity.

An algebraic expression composed partly of an entire

QUEST.-What is a fraction? What does the numerator show?

What does the denominator show!
What is an entire quantity? ·

quantity and partly of a fraction, is called a mixed

quantity.

PROBLEM I.

(79.) To reduce a fraction to lower terms.

The value of a fraction is not changed if we multiply or divide both numerator and denominator by the same number.

Hence, to reduce a fraction to lower terms, we have the following

RULE.

Divide both numerator and denominator by any quantity which will divide them both without a remainder.

If the numerator and denominator are both divided

QUEST.-What is a mixed quantity? How may we reduce a fraction to lower terms? Give the rule.

until they no longer have any common factor, it is eviIdent that the fraction will be reduced to its lowest terms. In the case of monomials, it is easy to detect the presence of a common factor; in the case of polynomials, they may often be detected by applying the principle of Art. 68.

QUEST.-How may we reduce a fraction to its lowest terms?