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.
$\left(x+2\right)\left(xy\right)$
${x}^{2}\mathit{\text{xy}}+2x  2y$
Where did the first term,
${x}^{2}$
, come from?It is the product of
$x\mathit{\text{ and }}x$
, the first terms in $\left(x+2\right)\text{and}\left(xy\right)$
.The next term,
$\mathit{\text{xy}}$
, is the product of $x\text{ and }y$
, the two outer terms.The third term,
$+2x$
, is the product of $2\text{ and }x$
, the two inner terms.And the last term,
$2y$
, came from multiplying the two last terms.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.
Let's look at
$\left(x+3\right)\left(x+7\right)$
again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.example
Multiply using the FOIL method:$\left(x+6\right)\left(x+9\right)$
.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.  $x^2+15x+54$ 
try it
146211We 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
 Multiply the First terms.
 Multiply the Outer terms.
 Multiply the Inner terms.
 Multiply the Last terms.
 Combine like terms, when possible.
example
Multiply:$\left(y  8\right)\left(y+6\right)$
.Show Solution
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  $y^22y48$ 
try it
[ohm_question]146212]example
Multiply:$\left(2a+3\right)\left(3a  1\right)$
.Show Solution
Solution
$(2a+3)(3a1)$ 

Multiply the First terms.  
Multiply the Outer terms.  
Multiply the Inner terms.  
Multiply the Last terms.  
Combine like terms.  $6a^2+7a3$ 
try it
[ohm_question]146213[/ohm_question]example
Multiply:$\left(5xy\right)\left(2x  7\right)$
.Show Solution
Solution
$(5xy)(2x7)$ 

Multiply the First terms.  
Multiply the Outer terms.  
Multiply the Inner terms.  
Multiply the Last terms.  
Combine like terms. There are none.  $10x^235x2xy\color{red}{+7y}$ 
try it
[ohm_question]146215[/ohm_question]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 twodigit numbers.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.
example
Multiply using the vertical method:$\left(5x  1\right)\left(2x  7\right)$
.Solution
It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied
$23\left(46\right)$
.Multiply $2x  7$ by $1$ . 

Multiply $2x  7$ by $5x$ . 

Add like terms. 
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.
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