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10268 英文中學會考試題預習專欄

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數學科 (十三) 歐陽鋊文

MATHEMATICS(E)

·LESSON 13 FACTORISAZIon:(conta

If two or more integral expressions give a product which equals to another given expression, them these integral Rxpressions are called factors of the given expression. And the process of finding factors is called resolution into

factors or factorisation

Resolution into factors in an «inverse" operation and is. difler hom the dinet operation of multiplication Any two or more s cxpressions can be multiplied together and their product can be easily found. But, in general » an algebranal expression written down of random, has no simple & factor 0.g. 32+4 cannot be expressed as the Simple Expressions (other than I ¢ itself), product of fore

Again, there is no general method of factorisation. a«multiplication, we have a definite process which always gives us the product. But in order to factor. ise a given expression.

we have to learn a number

י

of special devices. It should be noted that when one factor of an expression is known, the other factor(s) can also be obtained by division

As Expression in which each term has a monomial facía:

·à12x2 into factors Example 1 Resolve 3a" - sa'

SOLUTION Exp. =3a2-5a37*

ExAMBLE 2: Evaluate by using factors, & of th

SOLUTION:

+ 3 of 11036 + 466)

=₫ of 1500

= 600

<B> Expression in which the terms can be arranged in groups which have a common facta: EXAMPLE à : Factorise (a) acb=3)++ (5-b) (b) abt ac - bet + cd

SOLUTION: (a) Exp: = a(b-63-4rb-63

Cla-4(-5)

{b} Exp. = a(b+c)=d_[b+c)

Impossible to resolve m

HONG

factor

cil) As a general rule, it is better to selece, whenever possible, a letter which occurs only

to the first degree

<C> Expression in quadratic form: QX ̄+bX+C

Consider the identity, Cha+p}{kx+q}=@x#ba+E Then, from which, we find

a=bk b = he+

in case of factorising ax $b* *C

to replace by by too equivalent

Apx" such that:

necessary

terms hqxhq+ Ap=b

(i) chh){pq)m ac

It is advisible that you should systematically and Completely do your working (especially in this stage)

ŠIAMBLE 4: Factorise --

SOLUTION Exp === (42) −162 +rs) 5

15 4167

=-[422-108 −62 +15] |

15

the form of Perfect square A*±2 AB*8*={AIB

EXAMPLE 7: Factorise 4x2+ 49-252° -1223.

SOLUTION Exp = 1

— (az-39) - (52)a

«F> The sum or difference of two cubes: A3±B={A±8XI=AB÷8¬

EXAMPLE 8 Factorise

x2+ys

SOLUTION · Exp = {X*)* + ¢g¥3a

d

= (X*+4°)(X*" = x*y*

NOTE : for the expression, X”±ign. (a) if n = any positive integer

then

− y * = ( 2+ y){ X*"'+I

(b) if n = positive even infeger then, X”-y' = {Z+Y><X"~~

(C) if=n= positive odd integer

then, - *** 5* = (X+yxx?!

<B> By the Factor Theorem

general.

We have proved in "Remainder Theorem" that if the rational integral, function F(X) is divided by x-h then the remainder R= F(h). In particular, if R = P(A)= then F(x) is exactly divisible by x-h In other words, a-h is a factor of FOX

a

30+19+0-11-2

In more general, let F(X) be any rational integrat function, and if F(-2)=0, then Rath is factor of F(X), this is thusfor called the "factor The EXAMPLE 9 Factorise 30×3+192"-1, SOLUTION - Let F(x)=301 +192*~) As shown, F(£)=F{·÷)=R{®)=0. 4+3x+1, 38+1, 52-1 are factos of Fras and their product gives an exp in des

FC) = R(32+1)(3241) (SX-1) comparing one of the terms, say the

30 X = 4(=INSL)(T2)*

f(1) = (Ex+1)(3X+1)($X=1)

tenn

NOTE: 113 An expression eig. a3⁄4(b−c)+ b3(c=a]+

is unatter

a,b,c called

expres Fact

The following

any given algebrai

changing 4-6, 6-

expression wint of letters is

This kind of clarised by

1; Write down first the common fecter,

Make use of the

crence (or sua) of two squam

(or cubes), if possible. 3. It may be possible to factavise by grouping.

expression contains four or more terms,

if the expression contains brackets, and factor (or standard form) is seen should usually be removed and

is therefore Make

COMEM

the relation of possible, that

Fatbt es o

Use the facts Theorem and all the related properties coffenever required.

在三廿月一年八六九一帶会年会 五園民维

168英文中學會考試題預習專欄

英文科(十三)

MRS. BENNETT

Meldab '113)

Azgvera to Poser XII

1. People wond their sun windows.

2. Relativoiy hero maana wages for artisane vera

low before the var compared to their wares after the war.

3. Many

Available bere times the number of ekflled men still prepared to work sa artisana.

has learnt how to mend

to meet this need

the ordinalry una

A. 1. Wages for skilled vormas were low and 16-yna customary for the owner to have an artisan de his maintenance verk for him.

2. Many artisans were away fighting so owners bad

to do their own repairs.

Kary artisans found they vore paid higher wagca in factories co began working in them,

He objects to paying a skilled workmen a lot of songy to do something he has discovered be enjoys doing himself.

They feel they are so slever they want to make things as well. ⠀

vering the var, vnen many skilled man were fighting, maintenance of homes had to be done by owners, Suprisingly they discovered they enjoyed doing it and it grew into a habit.

After the war there was still a shortage of skilled workmen and householders continued doing their own repairs. "Do-it-yourself" kits and booke were sold to encourage the habit and the desire to

69 words. make new things spread rapidly,

Paper IIII

Put the verbs in bracketa into the correct tenseaz

I

(wait) in the coffee shop for a long

(promise) time for my friend Joan to come, She that she would meet me there at five o'clock, but #1 (bo) already half past six and she still (not come). I

(grov) impatient but 1

(think) my friend might have found difficulty

in getting on a bus. I

wait for another half hour.

"If she

(go home).

Just then I

the shop.

(decide) therefore ta

not turn up) by seven o'clock

(think).

(notice) my friend entering

(think) you

(maks) you so late?

two hours,

buăr my friend

(not come). What (wait) for

wait) for over an hour to get onva.

(reply). Then while ve der sala

·(be) an

come) along Queens Road there accident so that

KIES

(cause) another delay. I (not expect) to find you still here, but I think) I had better come just in case, Ve (miss) the beginning of the film but, if we

(hurry) va

(be) in time for most of. (make) you wait

It. I am very sorry that I

no long.

B. ll in the blanks with suitable prepositions,

i, azter much persuasion në gave;

majority.

che

the society

cruelty

2. I took the loht kitiem

the prevention

3. Although he began speaking

glance up

me.

animals. mo he did not

4. Glance

your composition again to check if yon have made any mistakes.

5. The teacher asked the students to go on

their work,

1

6. The society provides food and shelter

FOOT.

The poor will be providea by the society.

8. The child cannot get

they are locked up. ·

You can get away

can talk persuasively,

the

food and shelter

the medicines decausa

almost anything it you.

· (3-2x) (2x-5)

EXAMPLE 5. Factorise SOLUTION Exp

6X2 - X - 13

HINTS & ANS. TO EX. 12

f(-2) = 55;

F(26-3) ¤ [306=9)+4][a(at-3)=5]=(6t−5X4t~11)

2, GC -2)=16; _G(x+1,9+1)=(2-4)`s G(~29,-33)= g2.

3, X-5 (the remainder of the 1st degree 123

4 Let Flap = ax* - 21* + bx* - 6x-9

Then. R= F(-1)=a+b=1mo

R = F(3)=810+9b=2!=0

5,(2) Exp = (2-

X = y ] [ 3 x + y}< X99 y}{3X=29)

(b) No Morter form i NOTE :0)(df - e)" #-18+αs*

(c) (az - b)( bz-4a). +(e-a)2, when k is even

•ce-di”, when k is odd.

10. The second runner gradually gained.

leader.

C. Put the following into reported speech.

I beg your pardon, Mr. Johnson". "Hello." he said, turning round, "what's the

matter?"

"I wish tẹ be quite right, Hr. Johnson, and keep to your directions no I thought I had better. aak. Would there be any objection to ay taking leave of

yone, I know, about here before I go

he said, looking as ar he hardly

= ↓ ( Y −9Y+84-72)

={[Y(Y=9)+8¢Y-

1 factavise

EXERCISE 13

completely the following, if possible v

understood ma

-+(Y-9)(Y+8)

=4(61-9)(62+8)

NOTE: Some quadratic expressions can often be fact-

orised by inspection. Yet, the

answers should

always be checked mentally by multiplication, whichever what method is used. The above. example will be simple by using the "cross product" method

ATM- Bra(A+BXA-8)

Difference of two squares EXAMPLE GA Factorise 413x=xy)- €37+ SOLUTION | Exp. =[2(3x=2y}}' – (3x+2y)

cez

37x-

Find the two values of p for which the yaw ratic expressio

have

brand.

nity.

"I don't mean In the village only, but also the town

*. he said, "no objections:

Write one sentence for each of the following figurative expressions to show you understand the morning,"

1. to build castles in the air

2. to be tied to his mother's apron strings

3. to break the back of anything

4. to get one's back up.

-5, bo kayo a "bar"in, oriels bornet..

6. to have no backbone

7. A bad egz

8. off the bas

9. to be dead beat

10. in cold bloed