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144. Sometimes it is convenient to express a single fraction as a group of fractions. Example.

5x4y - 10xya +15y3 5xʻy 10.cy 1573
10x4y2 10x2y2 10xy? " 10x^ya

1 1 Зу

+

2y

MIXED EXPRESSIONS. 145. We

may

often express a fraction in an equivalent form, partly integral and partly fractional. It is then called a Mixed Expression.

2C+7

(x+2)+5 5 Example 1.

=1+
20+2
20+2

20+2 3x - 2 Example 2. 3 (20+5) - 15 – 2_3 (20+5) - 17 17

3
20+5
20+5

X + 5

X +5 In some cases actual division may be advisable.

22 – 7.0-1

4 Example 3. Show that

:=200 -1

2-3 By division

2 – 3 )2.c2 – 7:6 – 1(2.c – 1
2x2 - 6x

-1
2 + 3

- 4. Thus the quotient is 2.c – 1, and the remainder - 4. 2xc2 – 73 - 1

4 Therefore

= 2.0 -13

X-3

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146. If the numerator be of lower dimensions than the denominator, we may still perform the division, and express the result in a form which is partly integral and partly fractional.

23

5437 Example. Prove that = 23 – 6x3 + 18.05 1+ 3x2

1+ 3x2 By division

1+ 3x2 ) 2.0 (2x – 6x3 + 1835

2:x + 6x3

-0.x3
- 6x3 – 18x5

18205
1825 + 54x7

- 54x7 whence the result follows.

Here the division may be carried on to any number of terms in the quotient, and we can stop at any term we please by taking for our remainder the fraction whose numerator is the remainder last found, and whose denominator is the divisor, Thus, if we carried on the quotient to four terms, we should have 2x

162.9
=2x – 6x3 + 18.45 – 54x7 +.
1+3x2

1+ 3x2 The terms in the quotient may be fractional ; thus if x2 is divided by 203 – ?, the first four terms of the quotient are 1 a3 ab +

+

and the remainder is

a

+

x7

a12 x10

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2a

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147. Miscellaneous examples in multiplication and division occur which can be dealt with by the preceding rules for the reduction of fractions.

a?

2a2 Example. Multiply 0+ 2a

by 2.C - a 2x + 3a

x+a'

a
The product =
30 + 2a

2x
2.0 + 3a
2x2 + 7ax + 6a2-a?

2x2 + ax

- a? – 2a2 2x + 3a

3+ a 2.co + 7ax + 5a2

2x2 + ax

За?
2x + 3a

20 + a
(22C+5a) (a + a) (2.C + 3a) (x – a)
2x + 3a

X+a
=(2x + 5a) (2 - a).

Х

=

Х

Х

EXAMPLES XVI. b.

Express each of the following fractions as a group of simple fractions in lowest terms:

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a-6

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Perform the following divisions, giving the remainder after four terms in the quotient:

7. x=(1+x). 8. a:(a-). 9. (1+x)= (1 - 3). 10.1+(1-3+x"). 11. 2? :(x+3). 12. 1:(1-x)

ai 63

382 13. Show that

(a - b)2

=a+26+ 14. Show that 22 xy + y2

273

x+y x+y 60.23 – 17x2 - 4x + 1

49 15. Show that

12x – 25+ 5.7% + 9x – 2

x+2' 16. Show that 1+

a2+32 - (a+b+c) (a+b-c)
2ab

2ab
16x – 27

13 17. Divide

by x-1+
x2 – 16
16.x3

6x (a +4x) 18. Muitiply a’ – 2ax +4.x2

a + 2x

a2 + 2ax + 4x2 12

262 19. Divide 32+36-2 by 3b +6–

b-3

5-3 6564

1382 20. Divide a2 +962 + by a +36+

a? – 962

x+

X +4

by 3

a-36

98.x – 27 21. Multiply 4.22 +14x+

2x – 7

by

1
6

3x +29 12.02 +18x+27

148. We add a Miscellaneous Exercise in which most of the processes connected with fractions will be illustrated.

EXAMPLES XVI. C.

Simprify the following fractions :

1.

]

2.

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5.

2

6.

a? - ac a2 + 2ax + x2

Х
36 (c? – x2)

+
3 (+a) (x + 2a) (x+a) (2.c +a)
3a

6a
1
1

2

4.

x+y a2 + ab 262

X-Y) 2 2

4.2 2-1 +x+1-22-x+T: x2

2.34 (22+

1-28 1 1

2

+ (x+1)2

1 + x + x2 1 + x3

2.2.3 – 9x2 +27

9. 1 + 2x + 2.x2 + 203

3.x3 - 81x + 162 (a-12).x, a(a - b)..?

62 6-(b+ax) s xt - at

.206 - a2.23 •

x 122 – 2axta?

203+a

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7.

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x +1

8.

a

10.

+

x2 + axl

11,

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12.

13.

1+ a- x2

a a2 + ax + 22

a3 - 23 XA – 2x2 + 1

14.
3.25 - 10.203 + 15x – 8'

4
2
3

a
+
a at1 a+2 1

a+a(1+a) y+ya

a4 ya

15.

+

1+

a

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1 2x

27.

169 - a
62 +a)

2-1 3-2. (+) (-)

(6-4).

a

29.

.

a-6

a-6
b+
1+ab

1- ab
(a-6) a(a-6)

1
1+ab

I-ab
x2 + y2
y y
x2 - y2

31.
1 1 23 + 439
y

30.

4 2.0 +1

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