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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

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
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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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