8. x2-7x+12, 3x2-6x-9 and 2x3 6x2- 8x.

9.

8x3 + 27, 16x1 +36x2 +81 and 6x2 – 5x – 6.

10. x2-6xy + 9y2, x2 → xy — 6y2 and 3x2 – 12y3.

11. x2 - 7xy + 12y3, x3 – 6xy + 8y2 and x2 – 5xy + by3.

12. Shew that, if ax2 + bx + c and a'x2 + b'x + c' have a common factor of the form x+f, then will

(ac' — a'c)2 = (bc' — b’c) (ab′ — a′b).

13. Shew that, if ax3 + bx2 + cx+d and a'x3 + b'x2 + c'x + d' have a common quadratic factor, then will

14. Find the condition that ax3 + bx + c and a'x3 + b'x + c' may have a common factor of the form x+f.

15. If g1, 92, 93 are the highest common factors, and 11⁄2, 1, 11⁄2 the lowest common multiples of the three quantities a, b, c taken in pairs; prove that 9,99 (abc)3.

31

=

16. If A, B, C be any three algebraical expressions, and (BC), (CA), (AB) and (ABC) be respectively the highest common factors of B and C, C and A, A and B, and A, B and C; then the L. C. M. of A, B and C will be

A. B.C. (ABC) ÷ {(BC). (CA). (AB)}.

106.

CHAPTER VIII.

FRACTIONS.

WHEN the operation of division is indicated by placing the dividend over the divisor with a horizontal line between them, the quotient is called an algebraical fraction, the dividend and the divisor being called respectively the numerator and the denominator of the fraction. Thus means a ÷ b.

α

Since, by definition,

b

= a + b, it follows that z

xb=a.

b

107. Theorem. The value of a fraction is not altered by multiplying its numerator and denominator by the same quantity.

Hence

x x b x m = a × m;

:. x × (bm) = am. [Art. 29, (ii).]

Divide by bm, and we have

108. Since the value of a fraction is not altered by multiplying both the numerator and the denominator by the same quantity, it follows conversely that the value of a fraction is not altered by dividing both the numerator and the denominator by the same quantity.

a2x

Hence a fraction may be simplified by the rejection of any factor which is common to its numerator and denominator. For example, the fraction takes the b2x simpler form

when the factor x, which is common to its

numerator and denominator, is rejected.

When the numerator and denominator of a fraction have no common factors, the fraction is said to be in its lowest terms.

To reduce a fraction to its lowest terms we must divide its numerator and denominator by their H. C.F.; for we thus obtain an equivalent fraction whose numerator and denominator have no common factors.

The H.C. F. of the numerator and denominator is 3axy; and

Since x-a=- (ax), if we divide the numerator and denominator by a-x, we have the equivalent fraction ;

-X

a + x

numerator and denominator by x- -a, we have

-X

=

х

and if we divide the

Law of Signs in Division

a + x - (a + x)

The H. C.F. will be found to be x2-3x+7; and, dividing the numerator and denominator by x2 - 3x+7, we have the equivalent

109. Reduction of fractions to a common denominator. Since the value of a fraction is unaltered by multiplying its numerator and denominator by the same quantity, any number of fractions can be reduced to equivalent fractions all of which have the same denominator.

The process is as follows. First find the L.C. M. of all the denominators; then divide the L.C.M. by the denominator of one of the fractions, and multiply the numerator and denominator of that fraction by the quotient; and deal in a similar manner with all the other fractions: we thus obtain new fractions equal to the given fractions respectively, and all of which have the same denominator.

The L. C. M. of the denominators is x3y3 (x2-y2). Dividing this L.C. M. by x3y (x+y), xy3 (x − y) and x2y2 (x2- y2), we have the

quotients y2(x − y), x2 (x+y) and xy respectively. Hence the required fractions are

3

x3y (x+y)=x3y (x+y) × y2 (x − y) − x3y3 (x2 — y2) '

b

=

=

xy3 (xy) xy3 (x − y) × x2 (x+y) ̄ ̄x3y3 (x2 - y3)'

It is not necessary to take the lowest common multiple of the denominators, for any common multiple would answer the purpose; but by using the L. C. M. there is some saving of labour.

110. Addition of fractions. The sum (or difference) of two fractions which have the same denominator is a fraction whose numerator is the sum (or difference) of their numerators, and which has the common denominator. This follows from Art. 43.

When two fractions have not the same denominator, they must first be reduced to equivalent fractions which have the same denominator: their sum, or difference, will then be found by taking the sum, or difference, of their numerators, retaining the common denominator.

When more than two fractions are to be added, or when there are several fractions some of which are to be added and the others subtracted, the process is precisely the same. The fractions must first be reduced to a common denominator, and then the numerators of the reduced. fractions are added or subtracted as may be required.

The L.C.M. of the denominators is (a+b) (a − b); and