155. When the factors of the numerator and denominator cannot be determined by inspection, the fraction may be reduced to its lowest terms by dividing both numerator and denominator by the highest common factor, which may be found by the rules given in Chap. XVIII.

Example. Reduce to lowest terms

3x3 13x2+ 23x - 21
15x3-38x-2x+21

The H.C.F. of numerator and denominator is 3x-7.

Dividing numerator and denominator by 3x-7, we obtain as respective quotients x2 - 2x + 3 and 5x2-x-3.

Thus

3x3 - 13x2+23x − 21 (3x − 7)(x2 - 2x+3)_ x2 - 2x+3
15x3-38x2-2x+21 (3x-7)(5x2-x-3)5x2-x-3

156. If either numerator or denominator can readily be resolved into factors we may use the following method.

Example. Reduce to lowest terms

x3+3x2 - 4x 7x2-18x2+6x+5

The numerator = x(x2 + 3x − 4) = x(x+4)(x − 1).

Of these factors the only one which can be a common divisor is Hence, arranging the denominator so as to shew x-1 as a

x-1.

factor,

Multiplication and Division of Fractions.

157. Rule. To multiply together two or more fractions: multiply the numerators for a new numerator, and the denominators for a new denominator.

and so for any number of fractions.

In practice the application of this rule is modified by removing in the course of the work factors which are common to numerator and denominator.

by cancelling those factors which are common to both numerator and denominator.

158. Rule. To divide one fraction by another: invert the divisor, and proceed as in multiplication.

17.

18.

64a2b2 - 1 x2 - 49 x-7

x2-x-56 8a3b-a2 a2x-8a2

4x2+4x-15
x+8
x2+2x-48 2x2-15x+18 (x-6)2

a2+8ab-9b2a2 - 7ab + 12b2, a3 + a2b ab2

x2+ax - 20a2 x28ax+16a2 x2-ax-30a2 ax2+9a2x+20a3 x2+8ax+15a2*

(a - b)2 - c2 a2 + ab + ac ̧ (a+b)2 – c2

X

X

a2-ab+ac (a–c)2 - b2 (a+b+c)2

[For additional examples see Elementary Algebra.]

2x2+5x

÷

a3 - b3

X

a2-3ab-4b2"

CHAPTER XX.

LOWEST COMMON MULTIPLE.

159. DEFINITION. The lowest common multiple of two or more algebraical expressions is the expression of lowest dimensions which is divisible by each of them without remainder.

The lowest common multiple of compound expressions which are given as the product of factors, or which can be easily resolved into factors, can be readily found by inspection.

Example 1. The lowest common multiple of 6x2(a − x)2, 8a3(a − x)3, and 12ax(ax)5 is 24a3x2(a - x)3.

For it consists of the product of

(1) the L.C.M. of the numerical coefficients;

(2) the lowest power of each factor which is divisible by every power of that factor occurring in the given expressions.

Example 2. Find the lowest common multiple of 3a2+9ab, 2a3 - 18ab2, a3+6a2b+9ab2.

Resolving each expression into its factors, we have 3a2+9ab=3a(a+3b),

2a3 - 18ab22a(a + 3b)( a − 3b),

a3+6a2b+9ab2 = a(a+3b) (a+3b)
= a(a+3b)2.

Therefore the L.C.M. is 6a(a + 3b)2(a − 3b).

Example 3. Find the lowest common multiple of
(yz2 - xyz)2, y2(xz2 − x3), z1+2xz3 +x2z2.

Resolving each expression into its factors, we have
(yz2 — xyz)2 = {yz(z − x)}2 = y2z2(z − x)2,
y2(xz2 — x3) = y2x(~2 — x2) = xy2(z − x)(z+x),
4+2x-3+x2-2 = ≈2(~2 + 2xz + x2) = x2(z+x)2.

Therefore the L. C. M. is xy22(z+x)2(z − x)2,

22. 2x2-7x-4, 6x2 -7x-5, x3- 8x2+16x.

23. 10x2y2(x3 — y3), 15y1(x − y)3, 12x3y(x − y)(x2 — y2).

24. 2x2+x-6, 7x2+11x-6, (7x2 – 3x)2.

25. 6a3-7a2x - 3ax2, 10a2x - 11ax2-6x3, 10a2-21ax - 10x2.

160. When the given expressions are such that their factors cannot be determined by inspection, they must be resolved by finding the highest common factor.

Example. Find the lowest common multiple of

2x+x3- 20x2 - 7x+24 and 2x4+3x3- 13x2-7x+15.

The highest common factor is x2+2x-3.

By division, we obtain

2x4+x3- 20x2 - 7x+ 24 = (x2+2x-3)(2x2 - 3x-8).

2x4+3x3- 13x2 - 7x+15= (x2 + 2x − 3)(2x2 - x − 5).

Therefore the L.C.M. is (x2+2x − 3)(2x2 - 3x − 8)(2x2 -- x − 5).