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93. Signs in Fractions. There are three signs to be considered in a fraction : first, the sign of the numerator; second, the sign of the denominator; and third, the sign of the fraction itself, which is placed just before the dividing line. Thus, in the fraction
the sign of the numerator is –, the sign of the denominator is +, while the sign of the fraction itself is +. Again, in the fraction
the sign of the numerator is t, that of the denominator is –, and that of the fraction itself is – .
Since a fraction is merely an indicated division, the law of signs for division ($ 70) must hold for all fractions. Thus we have,
+-12 = +2;
From these examples it appears that if any two of the three signs of a fraction are changed, the value of the fraction is not changed.
Care must be taken, however, in changing the sign of the numerator or denominator of a fraction if polynomials are present. Thus, if the numerator is a polynomial, we can change the sign of the whole numerator only by changing the sign of every term in it. A similar statement applies when the denominator is a polynomial. For example, we have a+2 6+c -a-2b
c a +2 6+c *2ą-3 6–2 c 2 a-3 6-2 -2 a+3 6+2 c Observe carefully the reason for every change of sign here.
ORAL EXERCISES 1. State three other ways of writing the fraction
2. State three other ways of writing -
3. State three other ways of writing each of the following fractions : _2 9 -4 2 a _ a+b 2 x+y x2+x-1
c-d' 2-1 3 xy+7
Change the following into fractions having no negative signs in either their numerator or denominator.
-c-3 y 5. G 9. c 13. - - 3–5.
y+3 x+2y+z - 3 x2 — Y-22 -mn-m3 -n2 6 pq+r+s2
94. Reduction of Fractions to Lowest Terms. A fraction is reduced to its lowest terms when its numerator and denominator have no common factor except 1. Thus, each of the fractions
5 x 2 a
6'y' 36' " is in its lowest terms, but
, and a+b
a+b are not in their lowest terms. Explain why not in each case.
To reduce a fraction to its lowest terms, factor the numerator and denominator and then divide both numerator and denominator by all their common factors. EXAMPLE 1. Reduce 25 a 0°x to its lowest terms.
35 a’bx SOLUTION. 25 a2b3 x _5.5:4 8 . b. b. x_ 5 62.
. Ans. 35 a'bx 7.7.fe. Ja.a.%.& 7a Thus, division in algebra may be carried out by canceling, just as in arithmetic. Doing this is equivalent to dividing both numerator and denominator by all their common factors, which is the same as dividing them by their H. C. F. EXAMPLE 2. Reduce a“ -110724 to its lowest terms.
a’– 5 a+6 SOLUTION.
q?-11 a+24 _ (a-8)(0-3)_ a-8. Ans.
0 3) -2 .. Observe that only factors that are common to both numerator and denominator can be canceled. It is a common error on the part of students to cancel terms instead of factors. Thus, in the fraction
a+b' the a's cannot be canceled, for, though a is here a common term, it is not a common factor of numerator and denominator.
Another common error is to write 0 for the result whenever all the factors of numerator and denominator cancel each other. The answer in such cases is always 1 instead of 0. Thus
80) (to) The principle to be remembered here is that any number when divided by itself gives 1. Thus,
2-1, 4-1, -3=1, 3 =1, a+b=1, etc.
ORAL EXERCISES Reduce each of the following fractions to its lowest terms.
10 x_y2 36 xr3 72 xöp3 2 abc abc
7. 8+8 x+2 x2
20 x?y22 a 88 a6x4b2y
a2 — 62
- 150 xøy3 0. a? — 2 ab+62" 75 abcd
(a−b)2 o q2—62 150 a2b3c4d5
a? — 2 ab+62 ' a2+2 ab +62°
10 9x2 - y2__1 a2b2+ab
2 x2y—2 x2
3 cou-3 có 13. 4x2 — Y2.
y–2 x [Hint. Change the sign of the denominator and the sign of the fraction. Then the fraction takes the form - 4x2. Seeg 93.]
2 x-y 92—6 8+8
tea— 15 ab- 34 62 52–5 8+6
a?–6 ab- 16 62 16 a2b2+10 ab+21
12 m+ -man2 a2b2+11 ab+24
m2 — m2
p4 — 6 p2+5 * – 27 – 23
22—9 10 a?–a–2.
63+7 62-86 - a2+4
– p6+p2 [Hint. See Ex. 13.] *1 x2 - y2. 20 x2+x-6
x3 — y3 15+2 x — x2
– 23+27 72 — 2 rs-15 52 * x2 +6 x—27
–4 y2 4472 +8 rs+15 52 [Hint. See Ex. 13.]
- (r–2 y)2 For further exercises on this topic, see the review exercises, p. 161, and Appendix, p. 301.
95. Reduction of Fractions to Common Denominator. Such fractions as s, š, 131 are said to have a common denominator; that is, the denominator of each is the same, being 3. Similarly, the fractions
a 3 44 x 2
o o toto have the common denominator b; and the fractions
a 69 x2 + y2
a+b' a+b' a+b have the common denominator a+b.