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1. (a + b) (a – b) = a 2 – ab + ab – b 2 = a 2 – b 2 (vereenvoudig)

2. (a + 2) (a + 3) = a 2 + 3a + 2a + 6 = a 2 + 5a + 6

3. (a + b) (a + b) = a×a +ab + ba + b×b = a 2 + ab + ab + b 2 = a 2 + 2ab + b 2 (vereenvoudig)

4. (a + b) (c + d) = ac + ad + bc + bd (hierdie antwoord kan nie vereenvoudig word nie)

  • Die antwoord op die soort probleem in vraag 1 hierbo het die vorm van ‘n verskil van vierkante.
  • Die antwoorde op 2 en 3 is drieterme. Ons gaan nou probeer om drieterme te faktoriseer.
  • Die eerste feit om te onthou is dat nie alle drieterme gefaktoriseer kan word nie .

Werk agteruit deur probleem 2:

a 2 + 5a + 6 = a 2 + 3a + 2a + 6 = (a + 2) (a + 3).

  • So is dit duidelik waar die a 2 vandaan kom, en die 5a en die 6.

Faktoriseer nou a 2 + 7a + 12 = ( ) ( ) deur twee geskikte tweeterme in die twee paar hakies te skryf.

  • As jy die tweeterme in die hakies uitvermenigvuldig soos jy in aktiwiteit 2.2 geleer is, kan jy jou antwoord toets. Hou aan en toets telkens jou antwoorde tot jy seker is hoe om dit te doen. Doen dieselfde in die volgende oefeninge:
  • Elke drieterm het ‘n maat in die tweede kolom; soek hulle uit:

A. a 2 – 5a – 6 1. (x + 2)(x + 3)

B. a 2 – a – 6 2. (x – 2)(x + 3)

C. a 2 – 5a + 6 3. (x + 1)(x – 6)

D. a 2 + 7a + 6 4. (x – 2)(x – 3)

E. a 2 + 5a + 6 5. (x + 1)(x + 6)

F. a 2 + 5a – 6 6. (x – 1)(x + 6)

G. a 2 + a – 6 7. (x + 2)(x – 3)

H. a 2 – 7a + 6 8. (x – 1)(x – 6)

  • Faktoriseer nou die volgende drieterme op dieselfde manier. Die laaste twee is moeiliker as die eerste vier!
  1. a 2 + 3a + 2
  2. a 2 + a – 12
  3. a 2 – 4a + 3
  4. a 2 – 9a + 20
  5. a 2 + ab – 12b 2
  6. 2a 2 – 18a + 40

Aktiwiteit 4

Om faktorisering te gebruik in die vereenvoudiging van breuke, en in die optelling, vermenigvuldiging en deling van breuke

[lu 1.2, 1.6, 2.9]

A. Vereenvoudiging van algebraïese breuke

Twee van die volgende vier breuke kan vereenvoudig word, en twee nie. Watter twee kan?

2 + a 2 a size 12{ { {2+a} over {2 - a} } } {}

3 a + b a + b size 12{ { {3 left (a+b right )} over {a+b} } } {}

4 + x x + 4 size 12{ { {4+x} over {x+4} } } {}

a b c 2 b + c size 12{ { {a left (b - c right )} over {2 left (b+c right )} } } {}

Jy het seker nou al agtergekom dat dit baie moeite is om te faktoriseer. Hoekom doen ons dit?

  • Hierdie breuk kan nie vereenvoudig word soos dit staan nie: 6a 2 b 6b 2a 2 size 12{ { {6a rSup { size 8{2} } b - 6b} over {2a - 2} } } {} . Dis omdat ons nie terme mag kanselleer nie. As ons die som uitdrukkings na produk uitdrukkings kan verander (deur faktorisering) sal ons die faktore kan kanselleer, en sodoende klaar kan vereenvoudig.

6a 2 b – 6b = 6b(a 2 – 1) = 6b (a + 1) (a – 1) en 2a – 2 = 2(a – 1)

  • Dus is die motivering vir faktorisering die behoefte aan vereenvoudiging.

Dus: 6a 2 b 6b 2a 2 size 12{ { {6a rSup { size 8{2} } b - 6b} over {2a - 2} } } {} = 6b a + 1 a 1 2 a 1 size 12{ { {6b left (a+1 right ) left (a - 1 right )} over {2 left (a - 1 right )} } } {} = 3b a + 1 1 size 12{ { {3b left (a+1 right )} over {1} } } {} = 3b(a + 1) .

Dit is baie belangrik om volledig te faktoriseer.

Oefening:

Faktoriseer beide teller en noemer, en vereenvoudig:

1 12 a + 6b 2a + b size 12{ { {"12"a+6b} over {2a+b} } } {}

2 x 2 9 x + 3 size 12{ { {x rSup { size 8{2} } - 9} over {x+3} } } {}

3 2 a + 1 a 1 6 a + 1 2 size 12{ { {2 left (a+1 right ) left (a - 1 right )} over {6 left (a+1 right ) rSup { size 8{2} } } } } {}

4 5a 2 5 5a + 5 size 12{ { {5a rSup { size 8{2} } - 5} over {5a+5} } } {}

B. Vermenigvuldiging en deling van breuke

  • Die gewone reëls om breuke te vermenigvuldig en te deel bly steeds van toepassing. Bestudeer die volgende voorbeelde – let veral op na die faktorisering en kansellering.

4x 3 y 6y 2 ÷ xy 3x 2 × 2 xy 2 3x size 12{ { {4x rSup { size 8{3} } y} over {6y rSup { size 8{2} } } } div { { ital "xy"} over {3x rSup { size 8{2} } } } times { {2 ital "xy" rSup { size 8{2} } } over {3x} } } {} = 4x 3 y 6y 2 × 3x 2 xy × 2 xy 2 3x size 12{ { {4x rSup { size 8{3} } y} over {6y rSup { size 8{2} } } } times { {3x rSup { size 8{2} } } over { ital "xy"} } times { {2 ital "xy" rSup { size 8{2} } } over {3x} } } {} = 4x 4 3 size 12{ { {4x rSup { size 8{4} } } over {3} } } {}

a 2 9 2 × 1 4a 2 12 a size 12{ { {a rSup { size 8{2} } - 9} over {2} } times { {1} over {4a rSup { size 8{2} } - "12"a} } } {} = a + 3 a 3 2 × 1 4a a 3 size 12{ { { left (a+3 right ) left (a - 3 right )} over {2} } times { {1} over {4a left (a - 3 right )} } } {} = a + 3 8a size 12{ { { left (a+3 right )} over {8a} } } {}

3a + 6 5 ÷ a 2 4 10 size 12{ { {3a+6} over {5} } div { {a rSup { size 8{2} } - 4} over {"10"} } } {} = 3a + 6 5 × 10 a 2 4 size 12{ { {3a+6} over {5} } times { {"10"} over {a rSup { size 8{2} } - 4} } } {} = 3 a + 2 5 × 10 a + 2 a 2 size 12{ { {3 left (a+2 right )} over {5} } times { {"10"} over { left (a+2 right ) left (a - 2 right )} } } {} = 6 a 2 size 12{ { {6} over {a - 2} } } {}

Oefening:

Vereenvoudig:

1. 2 ab 2 b 3 c × 9 ac 2 4b ÷ 3 ac 2b 2 size 12{ { {2 ital "ab" rSup { size 8{2} } } over {b rSup { size 8{3} } c} } times { {9 ital "ac" rSup { size 8{2} } } over {4b} } div { {3 ital "ac"} over {2b rSup { size 8{2} } } } } {}

2. 2 a + 1 a 2 2 a 2 3 a + 3 × 9 a + 1 a + 3 2 4 a 2 ÷ 3 a + 1 a + 3 2 a 2 2 size 12{ { {2 left (a+1 right ) left (a - 2 right ) rSup { size 8{2} } } over { left (a - 2 right ) rSup { size 8{3} } left (a+3 right )} } times { {9 left (a+1 right ) left (a+3 right ) rSup { size 8{2} } } over {4 left (a - 2 right )} } div { {3 left (a+1 right ) left (a+3 right )} over {2 left (a - 2 right ) rSup { size 8{2} } } } } {}

3. 4a 2 + 8a 2b + 4 × 3 b 2 + 2 3a 2 + 6a size 12{ { {4a rSup { size 8{2} } +8a} over {2b+4} } times { {3 left (b rSup { size 8{2} } +2 right )} over {3a rSup { size 8{2} } +6a} } } {}

4. x 2 1 5x 5 ÷ x + 1 2 15 x + 15 size 12{ { {x rSup { size 8{2} } - 1} over {5x - 5} } div { { left (x+1 right ) rSup { size 8{2} } } over {"15"x+"15"} } } {}

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Wiskunde graad 9. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col11055/1.1
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