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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

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
hey
Giriraj
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
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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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