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A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written “partial product.” Beneath 138 is 92. Beside 92 is written “partial product.” Beneath 92 is a line and 1058. Beside 1058 is written “product.”

You start by multiplying 23 by 6 to get 138 .

Then you multiply 23 by 4 , lining up the partial product in the correct columns.

Last, you add the partial products.

Now we'll apply this same method to multiply two binomials.

Multiply using the vertical method: ( 5 x 1 ) ( 2 x 7 ) .

Solution

It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied 23 ( 46 ) .

.
Multiply 2 x 7 by −1 . .
Multiply 2 x 7 by 5 x . .
Add like terms. .

Notice the partial products are the same as the terms in the FOIL method.

On the left, 5x minus 1 times 2x minus 7 is shown. Below that is 10 x squared minus 35x minus 2x plus 7. The first two terms are in blue, the second two in red. Beneath that is 10 x squared minus 37x plus 7. On the right, a vertical multiplication problem is shown. 2xx minus 7 times 5x minus 1 is written with a line underneath. Beneath the line is a red negative 2x plus 7. Beneath that is 10 x squared minus 35 x in blue. Beneath that, there is another line. Beneath that line is 10 x squared minus 37x plus 7.

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Multiply using the vertical method: ( 4 m 9 ) ( 3 m 7 ) .

12 m 2 − 55 m + 63

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Multiply using the vertical method: ( 6 n 5 ) ( 7 n 2 ) .

42 n 2 − 47 n + 10

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We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.

Multiplying two binomials

To multiply binomials, use the:
  • Distributive Property
  • FOIL Method
  • Vertical Method
Remember, FOIL only works when multiplying two binomials.

Multiply a trinomial by a binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method . We first look at an example using the Distributive Property.

Multiply using the Distributive Property: ( x + 3 ) ( 2 x 2 5 x + 8 ) .

Solution

.
Distribute. .
Multiply. 2 x 3 5 x 2 + 8 x + 6 x 2 15 x + 24
Combine like terms. 2 x 3 + x 2 7 x + 24
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Multiply using the Distributive Property: ( y 1 ) ( y 2 7 y + 2 ) .

y 3 − 8 y 2 + 9 y − 2

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Multiply using the Distributive Property: ( x + 2 ) ( 3 x 2 4 x + 5 ) .

3 x 3 + 2 x 2 − 3 x + 10

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Now let's do this same multiplication using the Vertical Method.

Multiply using the Vertical Method: ( x + 3 ) ( 2 x 2 5 x + 8 ) .

Solution

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

.
Multiply ( 2 x 2 5 x + 8 ) by 3. .
Multiply ( 2 x 2 5 x + 8 ) by x . .
Add like terms. .
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Multiply using the Vertical Method: ( y 1 ) ( y 2 7 y + 2 ) .

y 3 − 8 y 2 + 9 y − 2

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Multiply using the Vertical Method: ( x + 2 ) ( 3 x 2 4 x + 5 ) .

3 x 3 + 2 x 2 − 3 x + 10

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Key concepts

  • Use the FOIL method for multiplying two binomials.
    Step 1. Multiply the First terms. .
    Step 2. Multiply the Outer terms.
    Step 3. Multiply the Inner terms.
    Step 4. Multiply the Last terms.
    Step 5. Combine like terms, when possible.
  • Multiplying Two Binomials: To multiply binomials, use the:
    • Distributive Property
    • FOIL Method
    • Vertical Method
  • Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the:
    • Distributive Property
    • Vertical Method

Practice makes perfect

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

15 ( r 24 )

15 r − 360

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−3 ( m + 11 )

−3 m − 33

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−8 ( z 5 )

−8 z + 40

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n ( n 2 3 n )

n 3 − 3 n 2

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12 x ( x 10 )

12 x 2 − 120 x

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−9 a ( 3 a + 5 )

−27 a 2 − 45 a

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6 x ( 4 x + y )

24 x 2 + 6 xy

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5 p ( 11 p 5 q )

55 p 2 − 25 pq

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3 ( v 2 + 10 v + 25 )

3 v 2 + 30 v + 75

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2 n ( 4 n 2 4 n + 1 )

8 n 3 − 8 n 2 + 2 n

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3 r ( 2 r 2 6 r + 2 )

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−8 y ( y 2 + 2 y 15 )

−8 y 3 − 16 y 2 + 120 y

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−5 m ( m 2 + 3 m 18 )

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5 q 3 ( q 2 2 q + 6 )

5 q 5 − 10 q 4 + 30 q 3

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9 r 3 ( r 2 3 r + 5 )

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−4 z 2 ( 3 z 2 + 12 z 1 )

−12 z 4 − 48 z 3 + 4 z 2

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−3 x 2 ( 7 x 2 + 10 x 1 )

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( 2 y 9 ) y

2 y 2 − 9 y

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( w 6 ) · 8

8 w − 48

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Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using different methods.

( x + 4 ) ( x + 6 )

x 2 + 10 x + 24

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( n + 12 ) ( n 3 )

n 2 + 9 n − 36

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( y + 8 ) ( y + 3 )

y 2 + 11 y + 24

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( a + 6 ) ( a + 16 )

a 2 + 22 a + 96

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( u 5 ) ( u 9 )

u 2 − 14 u + 45

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( z 10 ) ( z 22 )

z 2 − 32 z + 220

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( b 5 ) ( b 24 )

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( x 4 ) ( x + 7 )

x 2 + 3 x − 28

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( v + 12 ) ( v 5 )

v 2 + 7 v − 60

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( 6 n + 5 ) ( n + 1 )

6 n 2 + 11 n + 5

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( 2 m 9 ) ( 10 m + 1 )

20 m 2 − 88 m − 9

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( 5 r 4 ) ( 12 r + 1 )

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( 4 c 1 ) ( 4 c + 1 )

16 c 2 − 1

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( 8 n 1 ) ( 8 n + 1 )

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( 3 u 8 ) ( 5 u 14 )

15 u 2 − 82 u + 112

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( 2 q 5 ) ( 7 q 11 )

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( a + b ) ( 2 a + 3 b )

2 a 2 + 5 ab + 3 b 2

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( 5 x y ) ( x 4 )

5 x 2 − 20 x xy + 4 y

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( 4 z y ) ( z 6 )

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Multiply a Trinomial by a Binomial

In the following exercises, using either method.

( u + 4 ) ( u 2 + 3 u + 2 )

u 3 + 7 u 2 + 14 u + 8

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( x + 5 ) ( x 2 + 8 x + 3 )

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( a + 10 ) ( 3 a 2 + a 5 )

3 a 3 + 31 a 2 + 5 a − 50

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( n + 8 ) ( 4 n 2 + n 7 )

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( y 6 ) ( y 2 10 y + 9 )

y 3 − 16 y 2 + 69 y − 54

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( k 3 ) ( k 2 8 k + 7 )

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( 2 x + 1 ) ( x 2 5 x 6 )

2 x 3 − 9 x 2 − 17 x − 6

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( 5 v + 1 ) ( v 2 6 v 10 )

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Everyday math

Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 13 times 15 . Think of 13 as 10 + 3 and 15 as 10 + 5 .

  1. Multiply ( 10 + 3 ) ( 10 + 5 ) by the FOIL method.
  2. Multiply 13 · 15 without using a calculator.
  3. Which way is easier for you? Why?
  1. 195
  2. 195
  3. Answers will vary.
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Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17 . Think of 18 as 20 2 and 17 as 20 3 .

  1. Multiply ( 20 2 ) ( 20 3 ) by the FOIL method.
  2. Multiply 18 · 17 without using a calculator.
  3. Which way is easier for you? Why?
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Writing exercises

Which method do you prefer to use when multiplying two binomials—the Distributive Property, the FOIL method, or the Vertical Method? Why?

Answers will vary.

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Which method do you prefer to use when multiplying a trinomial by a binomial—the Distributive Property or the Vertical Method? Why?

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Questions & Answers

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
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Akash Reply
it is a goid question and i want to know the answer as well
Maciej
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Abigail
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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.
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what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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anybody can imagine what will be happen after 100 years from now in nano tech world
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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