Vertical and FOIL Methods for Multiplying Two Binomials

Learning Outcomes

  • Use the FOIL method to multiply two binomials
  • Use the vertical method to multiply two binomials


 

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms.

(x+2)(x−y)\left(x+2\right)\left(x-y\right)


x2−xy+2x−2y{x}^{2}-\mathit{\text{xy}}+2x - 2y


Where did the first term,
x2{x}^{2}
, come from?

It is the product of
x and xx\mathit{\text{ and }}x
, the first terms in
(x+2)and(x−y)\left(x+2\right)\text{and}\left(x-y\right)
.

Parentheses x plus 2 times parentheses x minus y is shown. There is a red arrow from the first x to the second. Beside this, The next term,
−xy-\mathit{\text{xy}}
, is the product of
x and −yx\text{ and }-y
, the two outer terms.

Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a red arrow from the first x to the y. Beside this, The third term,
+2x+2x
, is the product of
2 and x2\text{ and }x
, the two inner terms.

Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a red arrow from the 2 to the x. Below that, And the last term,
−2y-2y
, came from multiplying the two last terms.

Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a black arrow from the 2 to the x. There is a red arrow from the 2 to the y. Above that, We abbreviate "First, Outer, Inner, Last" as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.

Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner. Let's look at
(x+3)(x+7)\left(x+3\right)\left(x+7\right)
again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.

.  

example

Multiply using the FOIL method:
(x+6)(x+9)\left(x+6\right)\left(x+9\right)
.

Solution

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.
x2+15x+54x^2+15x+54
 

try it

146211

 

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

Use the FOIL method for multiplying two binomials

  1. Multiply the First terms.
  2. Multiply the Outer terms.
  3. Multiply the Inner terms.
  4. Multiply the Last terms.
  5. Combine like terms, when possible.Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.


 

example

Multiply:
(y−8)(y+6)\left(y - 8\right)\left(y+6\right)
.





 

try it

[ohm_question]146212]

 

example

Multiply:
(2a+3)(3a−1)\left(2a+3\right)\left(3a - 1\right)
.





 

try it

[ohm_question]146213[/ohm_question]

 

example

Multiply:
(5x−y)(2x−7)\left(5x-y\right)\left(2x - 7\right)
.





 

try it

[ohm_question]146215[/ohm_question]

For another example of using the FOIL method to multiply two binomials watch the next video.



Multiplying Two Binomials Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it works only for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written You start by multiplying
2323
by
66
to get
138138
.

Then you multiply
2323
by
44
, 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.

example

Multiply using the vertical method:
(5x−1)(2x−7)\left(5x - 1\right)\left(2x - 7\right)
.

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)23\left(46\right)
.

.
Multiply
2x−72x - 7
by
−1-1
.		 .
Multiply
2x−72x - 7
by
5x5x
.		 .
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.
 

try it

[ohm_question]146216[/ohm_question]

 

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.

 

Licenses and Attributions

More Study Resources for You

Multiply two binomials..docx

MATH 141 •  Lansing Community College

The FOIL Method.docx

ALGEBRA 1 OL •  Keystone High School

5-6a The FOIL method.docx

MATH 122 •  West Virginia University, Potomac State College

The FOIL method arises out of the distributive property.docx

MATH MISC •  University of South Carolina

Foil method.docx

MATH 564 •  Central High School

ALGB FOIL Method.pdf

MATH MISC •  University Of Georgia

Alternative to FOIL method.docx

MATH ALGEBRA •  Aldine H S

Multiplying Two Binomials.pdf

MATH 324 123 •  Dove Science Academy Okc

8-3_Multiplying_Binomials_NOTES.pptx

MATHEMATICS 102 •  Westlake High School, Atlanta

BINOMINAL EXPRESSION.pptx

MATH MISC •  Yale University

Multiplying Two Binomials.docx

APES 101 •  Martin County High School

algebra_milestone 2_8.png

BUS 123 •  Universitas Negeri Jakarta

Multiply a polynomial by a polynomial in order to use the foil method.docx

CS 2210/01 •  Lahore Grammar School

Screenshot_20210122-113159_22_01_2021_11_36

COLLEGE AL MAT101 •  Western Governors University

Screenshot_20210123-195656_23_01_2021_19_58

COLLEGE AL MAT101 •  Western Governors University

u2-cgallenge1-8.docx

ME MISC •  University of the People

Screenshot_20210124-110852_24_01_2021_11_09

COLLEGE AL MAT101 •  Western Governors University

math report ko.docx

MATH 10 •  Liceo de Cagayan University

u2-cgallenge1-7.docx

ME MISC •  University of the People

F19.odt

AA 1 •  No School

{AA4C7652-B70E-4FF9-8855-43551F58C26F}.png.jpg

ALGEBRA 107 •  University of Maryland, University College

2.PNG

MATH 341 •  University of Texas

{41015003-287C-4EAD-A3F0-AD043DC23176}.png.jpg

ALGEBRA 107 •  University of Maryland, University College

u2-cgallenge1-9.docx

MATH TRIGONOMET •  University of the People

Week 3 Discussion 1a

MAT 222 222 •  Ashford University

8.3 notes mult binomials.pdf

AA 1 •  No School

algebra 2 page 14.jpeg

MATH 101 •  Cathedral High School

{6BF9AD7A-9829-4C0F-A8DB-E1E08D5C0C03}.png.jpg

ALGEBRA 107 •  University of Maryland, University College

algh 8.7

MATH Intro Alge •  Old Bridge High

FA 8-3 22PDF Study_Guide_and_Intervention_Multiplying_Polynomials.pdf

MATH Algebra 1 •  Hilton High School

Show More