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We may now state the rule for finding the approximate value of a positive incommensurable root by Horner's Method.

Find the integral part of the root by Sturm's Theorem or method of Art. 573.

Transform the equation into another, each of whose roots shall be less by the integral part of the root.

If in this transformed equation the coefficient of the first power of the unknown quantity and the last term have the same sign, another figure of the root should be found by the method used to find the integral part of the root. If, however, the signs of these terms are unlike, divide the latter by the former, and the first figure of the quotient will be, approximately, the next figure of the root.

Transform the last equation into another whose roots shall be less by this approximate figure of the root found by division, and proceed as before to find another figure of the root.

592. It sometimes happens that the division of the last term of the first transformed equation by the coefficient of x in that equation gives a quotient greater than unity. In that case, as where the signs of these terms are alike, we obtain another figure of the root by the method used to obtain the integral part of the root.

593. If in any transformed equation after the first the signs of the last two terms are the same, the figure of the root used in making the transformation is too large and must be diminished until these terms have unlike signs.

594. If in any transformed equation the coefficient of the first power of the unknown quantity is zero, we may obtain the next figure of the root by using the coefficient of the second power of the unknown quantity as a divisor and taking the square root of the result.

595. Negative incommensurable roots may be found by transforming the equation into one whose roots shall be positive [Art. 551], and finding the corresponding root. This result with its sign changed will be the root required.

596. By Horner's Method we can find approximately any root of any number; for placing "a equal to w we have for solution the equation X"=a, or " – a=0.

EXAMPLES L. 1.

Compute the root which is situated between the given limits in the following equations : 1. 23+ 10x2 + 6x— 120=0; root between 2 and 3. 2. 23 – 2x–5=0; root between 2 and 3. 3. 24 — 2:03 +213 — 23=0; root between 1 and 2. 4. 23+*— 1000=0; root between 9 and 10. 5. 23+3c2 +3 — 100=0; root between 4 and 5. 6. 2x3 + 3x2 — 4X – 10=0; root between 1 and 2. 7. 23 — 46x2 – 36x+18=0; root between 0 and 1. 8. 23 + 30—3=0; root between 1 and 2. 9. 28+2x–20=0; root between 2 and 3. 10. 23+ 10x2 + 8x — 120=0; root between 2 and 3. 11. 3x3 +5x—40=0; root between 2 and 3. 12. 2+ - 12.2 + 12x — 3=0; root between – 3 and -4. 13. 25 — 4x4 + 7003 — 863=0; root between 4 and 5.

Find the real roots of the following equations : 14. 23 - 3x – 1=0.

15. 23 – 22x - 24=0. 16. 24 – 8x3 + 12x2+43—8=0. 17. 24 + x3+x2 + 3x – 100=0.

Find to four decimals, by Horner's Method, the value of the following: 18. yil.

19, 13. 20. 55. 21. 7.

MISCELLANEOUS EXAMPLES.

X

1. Simplify b-{b-(a+b)-[6-(6-a-1)]+2a}.
2. Find the sum of
a+b-2(c+d), b+c-3d+a) and c+d-4(a+b).

1
3. Multiply qx+3y

žy by x - y. 4. If x=6, y=4, 2=3, find the value of V2x + 3y +z. 5. Find the square of 2 - 3x + x2.

x +3 X-4 6. Solve +

2. X-1 6 7. Find the H.C.F. of a3 - 2a-4 and a3 - 42 -4.

2a 26 a? +62 8. Simplify +

a+b a--

3

y 9. Solve

+ :13
5 4
1
y

3
3

8 10. Two digits, which form a number, change places when 18

is added to the number, and the sum of the two numbers

thus formed is 44: find the digits. 11. If a=1, b= -2, c=3, d= -4, find the value of

a2b2+02c+d(a-6)

10a-(c+b)2 12. Subtract – 22 + y2 – 22 from the sum of 1 1 1 1

1 +3z,

and 3 13. Write down the cube of x+8y.

24 – yt vy 14. Simplify

.24+xy 24-44

x2 + y2^xy + y2^x
3
2

4x + 1 15. Solve = (2x 7)

(26 – 8) +4.
5
3

15 16. Find the H.C.F. and L.C. M. of

24 +23+2x – 4 and 203 + 3x2 – 4.

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17. Find the square root of 4a++9 (1 -- 2a) +3a? (7 – 4a).

xta 18. Solve

y=
2 3

a
+
2 3

x2 + a2
19. Simplify
xta

22 + ax 20. When 1 is added to the numerator and denominator of a

3 certain fraction the result is equal to subtracted from its numerator and denominator the result

is equal to 2: find the fraction. 21. Show that the sum of 12a +6b-c, - 7a-b+c and a+b+6c, is six times the sum of 25a + 136 - 8c, – 13a – 13b C,

and -lla+b+ 10c. 3

1 22. Divide 22 - xy + ya by –

16

q; and when 1 is

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23. Add together 18

9 6 | 3
3x
2y - 32

4
24

and 30

(2x 8

5 24. Find the factors of

(1) 10x2+79x – 8. (2) 729cô - .

2x – 1 5x + 3 4X – 118 25. Solve

35

11 26. Find the value of

(5a -36) (a-6) - 6{3a-c (4a-1)--(a+c)},

+

17

[ocr errors]

when a=0, b= 27. Find the H.C.F. of

7.2.3 – 10x2 – 70+10 and 2.03 – x2 – 2x+1. 28. Simplify

1.2.2 -7.cy +12y2 22-5xy +4yo

22 + 5xy + y2x4+xy - 2y 29. Solve

3abx + y= 907

4abx+3y=1705 30. Find the two times between 7 and 8 o'clock when the hands

of a watch are separated by 15 minutes.

31. If a=1, b= -2, c=3, d= -4, find the value of

d4-46+a'-W2+w+a+d.
1 1

1
.

4 33. Simplify by removing brackets a4-4a3-(Ca- 4a+1)

-[-2-{at-(-40% -6a' - 4a)} -(8a-1)]. 34. Find the remainder when 5.x4 – 7.03 +3.02 — x+8 is divided

by x – 4. 35. Simplify

22 + y2 XY- y2
24

y

-11 36. Solve

3

х

C
Х

202 XY

- 74

[ocr errors]

+y=18

y-13

2x +

= 29

4 37. Find the square root of 4.20 – 12.34+28x3 +922 – 42x+49. 38. Solye .006x — •491+ •723x= –.005. 39. Find the L.C. M. of 23+y3, 3x2 + 2xy y2 and 23 – x2y + xy2. 40. A bill of $12.50 is paid with quarters and half-dollars, and

twice the number of half-dollars exceeds three times

that of the quarters by 10; how many of each are used ? 41. Simplify

(a+b+c) - (a - b + c)2+(a+b - c)2-(-a+b+c)2. 42. Find the remainder when a4 - 3a3b + 2a-62 – 64 is divided

by a? - ab+262. 43. If a=0, b=l, c= -2, d=3, find the value of

(3abc-2bcd) Vabc-cbd +3. 44. Find an expression which will divide both 4x2+3x – 10 and 4.303+729 – 3x – 15 without remainder.

ab 1 1

a? 0 45. Simplify

20-12 1 1 a?

a: +62 ī

at

a-6

х

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