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Inleiding

In hierdie hoofstuk sal jy leer hoe om met algebraïese uitdrukkings te werk. Hersiening van vorige faktorisering en vermenigvuldiging van uitdrukkings sal dus nodig wees voordat die nuwe leerstof uitgebrei word vir Graad 10.

Hersiening van vorige werk

Die volgende behoort bekend te wees, maar ons gee 'n paar voorbeelde ter herinnering.

Dele van uitdrukkings

Wiskundige uitdrukkings is soos sinne en elke deel het 'n spesifieke naam. Jy behoort vertroud te wees met die volgende name wat die dele van wiskundige uitdrukkings beskryf.

a · x k + b · x + c m = 0 d · y p + e · y + f 0
Naam Voorbeelde (geskei deur kommas)
term a · x k , b · x , c m , d · y p , e · y , f
uitdrukking a · x k + b · x + c m , d · y p + e · y + f
koëffisiënte a , b , d , e
eksponent (of indeks) k , p
grondtal x , y , c
konstante a , b , c , d , e , f
veranderlike x , y
vergelyking a · x k + b · x + c m = 0
ongelykheid d · y p + e · y + f 0
binomiaal uitdrukking met twee terme
trinomiaal uitdrukking met drie terme

Produk van twee binomiale

'n Binomiaal is 'n wiskundige uitdrukking met twee terme, soos ( a x + b ) en ( c x + d ) . As hierdie twee binomiale vermenigvuldig word, is die volgende die resultaat:

( a · x + b ) ( c · x + d ) = ( a x ) ( c · x + d ) + b ( c · x + d ) = ( a x ) ( c x ) + ( a x ) d + b ( c x ) + b · d = a x 2 + x ( a d + b c ) + b d

Vind die produk van ( 3 x - 2 ) ( 5 x + 8 ) .

  1. ( 3 x - 2 ) ( 5 x + 8 ) = ( 3 x ) ( 5 x ) + ( 3 x ) ( 8 ) + ( - 2 ) ( 5 x ) + ( - 2 ) ( 8 ) = 15 x 2 + 24 x - 10 x - 16 = 15 x 2 + 14 x - 16
    .

Die produk van twee identiese binomiale, is bekend as die kwadraat (of vierkant) van binomiale en word geskryf as:

( a x + b ) 2 = a 2 x 2 + 2 a b x + b 2

Gestel die twee terme is a x + b en a x - b , dan is hulle produk:

( a x + b ) ( a x - b ) = a 2 x 2 - b 2

Dit staan bekend as die verskil van twee kwadrate (of vierkante) .

Faktorisering

Faktorisering is die omgekeerde proses van die uitbreiding van hakies. Byvoorbeeld, as hakies uitgebrei word, word 2 ( x + 1 ) geskryf as 2 x + 2 . Faktorisering sal dus begin met 2 x + 2 en eindig met 2 ( x + 1 ) . In vorige grade het ons gefaktoriseer deur die uithaal van gemeenskaplike faktore en die verskil tussen twee vierkante.

Gemeenskaplike faktore

Faktorisering deur die uithaal van gemeenskaplike faktore, is gebaseer daarop dat daar faktore is wat in al die terme voorkom. Byvoorbeeld, 2 x - 6 x 2 kan as volg gefaktoriseer word:

2 x - 6 x 2 = 2 x ( 1 - 3 x )

Ondersoek: gemeenskaplike faktore

Vind die grootste gemene faktore van die volgende pare terme:

(a) 6 y ; 18 x (b) 12 m n ; 8 n (c) 3 s t ; 4 s u (d) 18 k l ; 9 k p (e) a b c ; a c
(f) 2 x y ; 4 x y z (g) 3 u v ; 6 u (h) 9 x y ; 15 x z (i) 24 x y z ; 16 y z (j) 3 m ; 45 n

Verskil van twee kwadrate

Ons het gesien dat:

( a x + b ) ( a x - b ) = a 2 x 2 - b 2

In [link] dui die = teken aan dat die twee kante altyd gelyk sal wees. Dit beteken dat 'n uitdrukking in die vorm:

a 2 x 2 - b 2

gefaktoriseer kan word as:

( a x + b ) ( a x - b )

Dus,

a 2 x 2 - b 2 = ( a x + b ) ( a x - b )

Byvoorbeeld, x 2 - 16 kan geskryf word as ( x 2 - 4 2 ) wat die verskil is tussen twee kwadrate. Dus, die faktore van x 2 - 16 is ( x - 4 ) en ( x + 4 ) .

Faktoriseer volledig: b 2 y 5 - 3 a b y 3

  1. b 2 y 5 - 3 a b y 3 = b y 3 ( b y 2 - 3 a )

Faktoriseer volledig: 3 a ( a - 4 ) - 7 ( a - 4 )


  1. ( a - 4 ) is die gemene faktor
    3 a ( a - 4 ) - 7 ( a - 4 ) = ( a - 4 ) ( 3 a - 7 )

Faktoriseer 5 ( a - 2 ) - b ( 2 - a )

  1. 5 ( a - 2 ) - b ( 2 - a ) = 5 ( a - 2 ) - [ - b ( a - 2 ) ] = 5 ( a - 2 ) + b ( a - 2 ) = ( a - 2 ) ( 5 + b )

Hersien

  1. Vind die produkte / Verwyder die hakies:
    (a) 2 y ( y + 4 ) (b) ( y + 5 ) ( y + 2 ) (c) ( y + 2 ) ( 2 y + 1 )
    (d) ( y + 8 ) ( y + 4 ) (e) ( 2 y + 9 ) ( 3 y + 1 ) (f) ( 3 y - 2 ) ( y + 6 )


  2. Faktoriseer:
    1. 2 l + 2 w
    2. 12 x + 32 y
    3. 6 x 2 + 2 x + 10 x 3
    4. 2 x y 2 + x y 2 z + 3 x y
    5. - 2 a b 2 - 4 a 2 b


  3. Faktoriseer volledig:
    (a) 7 a + 4 (b) 20 a - 10 (c) 18 a b - 3 b c
    (d) 12 k j + 18 k q (e) 16 k 2 - 4 k (f) 3 a 2 + 6 a - 18
    (g) - 6 a - 24 (h) - 2 a b - 8 a (i) 24 k j - 16 k 2 j
    (j) - a 2 b - b 2 a (k) 12 k 2 j + 24 k 2 j 2 (l) 72 b 2 q - 18 b 3 q 2
    (m) 4 ( y - 3 ) + k ( 3 - y ) (n) a ( a - 1 ) - 5 ( a - 1 ) (o) b m ( b + 4 ) - 6 m ( b + 4 )
    (p) a 2 ( a + 7 ) + a ( a + 7 ) (q) 3 b ( b - 4 ) - 7 ( 4 - b ) (r) a 2 b 2 c 2 - 1


Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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