1970-01-20 — Page 22

二期星日十二月一年〇七九一屆公年九五國民華中 資教僑華

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2. (B) an

數學科

3. (D) off

(十二)

4. (D) over

(A) up with

LATHEMATICS (12)

(A) up

8.

Facterisation: Remainder Theroram

Example 1: Divide 2x3 3x + 5x - 7 by x- 2. What

is the remainder? What is the value of the divided when I - 21.

We find the quotient by using the method of detached boefficients

(i) (2x3 - 3x2 + 5x − 7) − (1 − 2)

212

» 2x + x.+ 7 ........... remainder 7

(11) Lot F(x) = x2

then x = 2

3x+5x=7

((2) 2 x 23-3x22 + 5 x 2 − `

- 16 - 12 + 10 = 7 - 7

From which we find that the remainder of

F(x) = (x-2) is the same as F(2)

The Remainder Theorem states that, if P(x) ve retional, integral, algebraical function of 1, th remainder on dividing 7(x) by ax + b is pla

a, b are constants and ax + b fo

where

When F(x) is divided by (ax + b) let the quotiens (x) and the remainder R. Then it is krown from

the ordinary rules of division in algebra that R will be a constant and will not contain x. Henos F(x)=(ax + b) Ø (=) + R

Teing I

b

W

A

in this identity

f(-)-0x

- 0 x Ø (~ 2-3) + 8&

.*. - f(--

Note: A rational, integral, algebraical function. x is an expression in x im which the power of x present are politive integers and the coefficiente of terms in I are all rational,

Example 2 Find the remainder on dividing

2x + 3) by (2x+

DI

(3x3

Let F(x)=3x3 - 2x + 3

Therefore required remainder

F(-1) 3(-7)3

-

3 + 1 + 3 -

2(-4) + 3

36

ENGLISH LANGUAGE (12)

Anawaza to BESTzino 11

1. (4) put you right

(C) to pisaCI

(C) brains

9. (4) newD7

0. (3) morals

.1. (A) lengalebing

12: (C) god-fozsaksa

13. (4) erfundva

14. (4) dimish

Exercise 12

Precio

15. (A) cockpit

Cramory

16.

17. A) prestige

18. (C) would not havs boca

19. (8) would retuwa

20. (e) had

21. (0) had know

22. (3) nor did

23.

24.. (0) It being fina

25. (B) wounded

25. (3) studied

27. (3) shall bam finished 28. (5) Having kroksa the door

Foad the following passage carefully i

Bad manners ara alvega objectionchle. Ea ora emonsa iguorant and badly brought up people, ubo know no better; but there are many, who do know better, who pride themselves on being rule and off-hand; and these vo cannet erowes. Some of these people (conceitqu follows!) think that rudeness is sigh of indapondencs and manliness, fad others say that politeness 19 a form of insincerity, and hold that to say that you are glad to see a person whom you really dislike, or that you are sorry wine a visitor has to leave when all the ties you are pind to get rid of bla, or to aak after a pozaen's health when you do not care whether he is alivs or dead, is simply hypocrisy. There may be aosathing in this objection; and yet a little innocent protonos and a few mild "social lins, may be lecz mozally wrong than the unnecessary wounding of paepla'a feolinga, Evon if you do not liza a man, ié is not always necessary to tell him the, brutal truth.

Good manners may be superficiel, and sometimes thay may be a little insincere; but they are as noomseary to the continuance of society as oil is to the working of a machine without friction, and with people who have naturally kind heɛrta, politaneas is net har insdześko nor artificial. For the 6690109 good manners in consideratize for the feelings of others and surely this is a virtue. Some ons, hés called good maszowe "surface religion", because the omenon of 1200 religion is unselfish ayapathy. You cannot like or approve of all you meets but you can and must learn to be kind to all, even to your Gnarios; and the least you can do is to treat than ocurisowly. There is enough rozraw in the world - without our trying to increase it by unnecessary umkindmann, which is the omnence of bad saaners, kat it costs a little to be polite – showing respect without servility to our superiors, courtesy to ou equals, and consideration for these belom us.

Ing tzio Cantlonna is one who instinctively ihlake of the feeling, the contest and heppingza of

there, before his con. En will therefore be courtcons to all. And there are teme gentlemen naturala

gentlemen, even amongst the poor end ignorant, who though they have not had the chance of learning al the rules of abiquette, have' kind heurts, question: Mako, a summary of the phasage explaining

people'a oyision on good mannera. Comprehension

A repayan, if the word in to be tekor i Ats literal Benso, is a paper which gives nova; end shio was all that the first nova papers undertook or attempted to do. But the daily prass 18 nov uzed for many other purscool. Bouidos giving news 17 €1753 advice, criticien, praias, or blaze, and in sprendi other respinta kas guge far beyond,1to original functions. Nowapapezu, a нe has know them, are, the organs of pačšia opinion on all kinds of news, 1ccsi', províncial, national, imperial, and foreign, and on all kinds of subjosun, anetal, political, legal, induntsåçl @cientific, litasams. The definition here given is a vide one, but anything less vide hould not exfer the ground. The phones "ongen of public opinion" presuppoži that the preus da fuse, - subject to Me Dazzorship End allowed, ne leng as it keeps itself hosest, unbribed, and wajibellous, to zus its courzO unimpeded in any okosnai or channels that it may select. In coupipies where the pressais not Szae, nous papers do not express the feeling of the nation,' but meraly 1d will of the sovereign or of those who Gxercise the Feuer of a moversiga. Of such newspapeED no sooount will be taken in the present owlay, Choose the best assmor that completes sack af they following: A

1. A noumpapar ás

(A) a word in livṣṭal Benem

(3) merely a papëz

(C) an attempt to do something

(D) a paper giving nowa

2. The new paper is. importeat be pauze

(A) it is a daily press

(2) 11 givas nagh'..

it bao ppnar far beyond itd original functione it is the organ of public opinion

3. A newspaper cas naver be the organ of publis

opinion if

(▲) it gives eli kanda of raus on all kinds of

subjects,

(B) 1+ subjects to no sensorship (G), the press is not froe

(D).1* doen met oorow a ́side ground

þ. á novapapez ean be a good' paper if it enjoys 329

freedom of press and

(A) it ocvana a wide' range of enbjosta

(3) it is from to choose ita chansol er channelą (c) it is met uabzibed

it is honest and walibellous

5. A necajaper is only expressing the vill of tha

sovereign. 1P

(A) it deos pot express' tho feeling of the ratios (B) itia madmpeded in any ohannal or chauffeża (c) thus is strict censorsháp

(D) it has, ne povez,

The Factor Theofem. This states that, if (ax + 3) ne a factor of F(1), then F(- 2) - 0, and conversely, if

(-) 0, then (ax + b) is a factor of f(x).

If f(x) has (*x + b) as a factor, then the remainder R obtained by dividing f(x) by (ax + be zero, and it has been shown that

(a -)3 - #

-

b) must

#0

if (ax + b) ba a factor of f(x).

Also if f(-) then R = 0 and (ax + b) must be

f(x)

a factor of

Similarly, if (z -α) be a factor of f(x), then (d)=

- 0, and, if f(x) - 0, then (x - K) is a factor of f(x)

-

←

xample 3 Without actual division, find the remainde a dividing 2x3 3x2 + 4x + 35 by (m)x + 2; (b) 2x -

Lot F(x)= 2x3 - 3x2 + 4x + 35

(a) Divide F(x) by x + 2. By Remainder Theorem

the remainder. - -(-2)

since f(x) is divisible by (x − 3), it follows that f(3) ww 24 + 3D -

from which p -8

=

Example Determine 'a' and '! in order that

+ 5x + b

3x + + x3

+

3x4 + x3 ax2 F(1) = 3(1)4 + .(1)3

may be exactly divisible both by x − 1 and x + 2

Let F(x)

+ 5x + b

Then R-

+

3 + 1 + a + 5 + b. a 0

0

a + b + 9

And Ra

F(-2)

付

(-2)+

́a(1) + 5(1), + b

(1)

+

(-2)3 a(-2)2 + 5(-2)

+ b

... 4a + b + 30 = 0 .. lat(2) – Bqt(1)

.....

(2)

3a + 21. 0

2

-7

43 - 8 + 48 - 10+ b = 0

by subutitution

-2

Example 6 Find the factors of

(1) x

(ii) x4.

-2(-2)

· 3(−2) + 4(−2) + 15 -16-12 - 8 + 35

--1

(b) Divide F(x) by 2x - 1

the remainder =

F(+7)

· 2(3) 3 − 3(†) + 4 (3\

+ − 2 + 2 + 35

36

Example 4 For what value

divisible by (x 3)?

Let f(x)=13 + * - -

\*• f(3) = 27 + 3P 3 - 24 + 31

H

- 14x + 24

+

- 3x + 2

Note. This means factorise an fully as possible) i) Considering the constant term 24, the only

possible factors are 1, 22, +3, +4, ±6, +8, +12, 24, and hence the only po-sible factors of the given function f(x) are (x + 1), (x + 2).. (x+3), (x + 4), (≈ ±6), (x+8),T(x ± 12) (x + 24). (Lowest values used first)

Lot f(x)=x3

.. (1)

- 14x + 24.

- - 1 - 14+ 240

.. (x-1) is not a factor of f(x)

28 + 24 0

2) is a factor of f(x)

f(2)

- 8 - 4

www.

(x

-

Dividing x

2

This gives

I

+ x - 12

-14x + 24 by x - 2

f(x)

=

(11) Let

(x − 2) (x2 + x −12)`

• (x − 2) (x − 3) (x + 4) f(x) ====+- 3x3 + 3x2 3x + 2

The only factors of 2 ara ±1, ± 2 f(1)

- 1 - 3 + 3 - 3 + 2 0

(x -1) is a factor of el«)

f(−1) = 1 + 3 + 3 + 3 + 240 ..(x-1) is not a factor of f(x)

f(2) 16- 24 + 12 - 6 ÷ 2 =

.. (x-2) is a factor of f(x) Dividing 14 3x3 + 3x2 gives x + 1

3x + 2 by (x - 1) (x − 2)

factors of f(x) are (x − 1)(x − 2) (x2 + 1} Note. (x2 + 1) cannot be expressed as the product

of roal factors and is known as factor.

a quadratic

Example 7 Find the roots of the equation

2x3 + 3x2

3x - 2 = 0

2x3

+ 3x

-

3x 2

Let f(x) Now f(1) 2 + 3 - 3 - 2 = 0

2x3

+

3x2

•'• (x-1) is a factor of f(x)

- 3x - 2 =

(2x3 - 2x2) + (5x2 - 3x − 2) (adding and subtracting 2x2) 2x2 (x − 1) + (x − 1) (5x + 2} (x = 1)(2x2 + 5x + 2)

ww

(x − 1)(x + 1) (x + 2) ·

Note: This method which replaces the division

method, makes use of the fact that (-1) is a factor of (x), therefore the given

équation can be written

(x-1)(2x + 1)(x + 2) = 0

x 1, or -2