Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

From the preceding rules are deduced the following useful theorems, viz :

1. By the rule for addition, if the sum of any two quantities, a and b, be added to their difference, the sum will be twice the greater. *

2. By the rule for subtraction, if the difference of any two quantities be taken from their sum, the remainder will be twice the less.t

3. By multiplication, example 2, article 21, if the sum of any two quantities be multiplied by their difference, the product will be the difference of their squares.

[merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][ocr errors][subsumed][ocr errors][merged small][subsumed][ocr errors][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small]

259

a

# Now, it is easy to perceive that the next, or sixth term of the 25

26 quotient will be and the seventh term and so on, alter

28 nately plus and minus; this is called the law of continuation of the series.

And the sum of all the terms, when infinitely continued, is said to be equal to the fraction att Thus we say the

2 vulgar fraction

3

when reduced to a decimal, is =.66666, &c. infinitely continued. The terms in the quotient are found by dividing the remainders by a, the first term of the divisor; thus, the first remainder, —X, divided by a, gives the second term in

poate the quotient; and the second remainder,+ divided by a, gives + the third term, &c.

2

a

EXAMPLES FOR PRACTICE. 1. Divide x_4832+200 by x+2. Ans. 22—50x+100. 2. Divide 28–222—24 by 3+4. Ans. 24–43—6. 3. Divide 20-49x2+4x480 by x+-5. Ans. 22–4x—16. 4. Divide x +39x4+249x+289 by x+1. Ans. 2—40x+289. 5. Divide 24_3822 210x +538x+289 by x+1.

Answer 208_39**+249x+289. 6

72 73 84
a+b) a (士十士
(1== ta ta ta, &c. answer.

&c
a+b

20 36? 483 a? +2ab+6?)a?....(1– +

um a 7? Fb.

a+2ab+82

a

[ocr errors]

a

a

[ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

2...

) الي

10

[ocr errors]
[ocr errors]

1

2x2 2x4

2x8

22.10 a?+°)a?—241– +

+

+ &c. a? at as as a'zo -2.24 am_2^)am_xmamat ntaminen, +&c. 224

am

"х"
-2x?
Q?

ta
2x4

-amxra ngan
+
a

+
2.24 2.26

ngan
ti
a

Rem. tammingen
2.26
at
226 228

mangan

man a

[ocr errors]

a

ta

[ocr errors]
[ocr errors]

Divisor.

b

at.2)+(5

axt

+

a art

a

[ocr errors]

2

ot

[ocr errors]

a

a

[ocr errors]
[ocr errors]

at

-&c.

a

[ocr errors]

a

a

[ocr errors][ocr errors][merged small][ocr errors]

Quotient.
bc bca bc bz+

20 .za 23 24. 6
&c.
a5
(1-4t

a bx

a? a b a>b2 a263 tb) a C +

&c. answer.

x2 203
bx
ba?

282 283 284 a+b)2ab(26 +

a a
bx

2ab+262
bx2
ba?

Quotient.

262
a?
bx2 ba? a 52

262
+
za

263
bx?
bx3 ab2

+
+
a?
+
x2

203 284
bX3

a>b2

+
23
2c3

264
6.23 bx4
a73

a?
a*

264 255 bat a 53 a’84

a al 203

285 bxt, bz brs

a? 十一十 R.

az a5

6.26 Remainder

as

[merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

6. Divide 206+6x*—10x2_11222_2070—110 by x+5.

Answer 24+23_1522—372—22. 7. Divide 2+x2–152—372—22 by x+2.

Answer 2_-22-132—11, and x?_2x11. 8. Divide 23_2_131–11 by x+1. Ans. x-2x-11.

? 9. Divide x+6x4—10.08—112x—207x—110 by 2+2.

Ans. +4.23_18x?_763--55. 10. Divide x4+4x3—182?_-762—55 by x+5.

Ans. 23-22-132—11. 11. Divide 25+6x9—10x2—112x?—207x--110 by x+1.

Ans. x4 +5X8_15x2__97x—110. 12. Divide x +5.x8—15x?—972--110 by 2+2 and by <+-5.

Ans. 2+3x?_212—55. 13. Divide 4- 192+123x?_302.2-7-200 by 2--4.

Ans. 23—15** +63x+50. 14. Divide x4—2728 +262x2+3562~1200 by x+3.

Answer 20—30x2 +252x400.

1

ON THE REDUCTION OF ALGEBRAIC FRACTIONS.

X

a

[ocr errors]

a

a

a

and y

(42.) The rules managing algebraic fractions being of the same nature as vulgar fractions in common arithmetic, the operations are performed exactly in the same manner.

CASE I. To reduce a mixed quantity to an improper fraction.

RULE. Multiply the integer, or whole part, by the denominator of the fraction, and to the product add the numerator ; then under their sum place the original denominator.

b 1. Reduce at, and a peach to improper fractions.

. 6 axatb_a2+6

abx at

Ans.

Ans.

b Зr -5

72 2. Reduce 42

and 4ax—

to improper fractions. 4

2ax
3x-5 16x43x+5 13x+5
Here 4.2-

Ans.
4
4

4
12 4ax X 2ax-62

-72 8a'r and 4ax

Ans.
2ax
2ax

2ax a2-x2c 62y—d 3. Reduce im.

to improper fractions.

y
a2-x2 xatto 2x—ato

2yb2+d
Ans.

32-6 ax+22 4. Reduce 50

and x

to improper fractions.

2a 32-6

522-(3x46) 54–3x+b Here 50

Ans.
a
axta 2ax-ax+x)

Ans.
2a
2a


2x+5

2.1-5 5. Reduce 52

and 52- to improper fractions. 4

4 2.4+5 5x X4(2.4+5 20x-2.2-5 181-5 52

Ans. 4 4

4

4 2x–5 5xX4—(2x–5)

202—2x+5_18x+5 An. and 5x4 4

4

4 a- -B 6. Reduce a–+

to an improper fraction. a2-ax-b ax—ta?-axa2-x2b 24+ ++

Ans.

[ocr errors]
[ocr errors]

2

2

a

a

а

ax-2

and 1

[ocr errors]

-ax

[ocr errors]
[ocr errors]

and 2-y

y-5x

[ocr errors]
[ocr errors]
[ocr errors]

a

a

3.c-4y—2 7. Reduce 2+3.x~

to improper 4.x

5 12x+7x+6 2xy2 fractions. Ans.

Ans. 4.2

5 5x

2x-5 8. Reduce 1- xt and 3.x— to improper fracy

5

bat 13x+6 tions. Ans.

Ans.
and

Ans.
Y

6

5
2x–8
Xma-1

2-3 9. Reduce 5+

1--
and 1+2x-

each

172+8 20–2+1 10x2+42+3 to improper fractions. Ans.

3x

5x 50(1+2x)-(x-3 ) 5x+102–3+3 1+2r5 5x

5x 102 +4x+3

Ans.

5x 2-a-1 2—(2--21) a—+a+1 2a—+1 1

Ans. a

a 3 3a

42-18

12+ to im4x

52 proper fractions. 1-Зас

238_3a 11. Reduce +

and 4+2x

to an improper fraction. (587)+3 38

3a 1xx За

X-3a
Ans.

Ans.
7
2ax 4x Baxta_8ax—3axa? 5axa?

Ans. 4.x

4x 64x-13 cx+1-3a-c 20a+10ax-2x +3a Ans. Ans.

Ans. 510

5a

а

a

10. Reduce 5-7, 1–

2a_3axta

[ocr errors]

9

с

5a

1

[ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

CASE II. To reduce an improper fraction to a whole, or mixed quantity. This is the same as Case II. in division.

Rule. Divide the numerator by the nominator for the integral part ; and if there be a remainder, place it over the denominator for the fractional part, with the proper sign prefixed.

ab_2a at 10.x?_52 +3 1. Reduce

to whole quanab a士

53 tities. abr2a? 2a 10x25x+3

3 Ans. ab 6 5x

52

[merged small][ocr errors]
« ΠροηγούμενηΣυνέχεια »