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Multiplying Polynomials #2 - The Foil Method

Introduction

Multiplying algebraic expressions is relatively easy when one of the expressions is a monomial.  Simply use the distributive property.

Multiplying_Polynomials_2_visual_1Multiplying_Polynomials_2_visual_2

 

However, the multiplication becomes more difficult when multiplying larger expressions such as binomials, trinomials, and larger polynomials.  The diagrams below give a visual example of the individual multiplications that must be done to complete these types of problems.

Multiplying_Polynomials_2_visual_3Multiplying_Polynomials_2_visual_4


 

Two binomials are multiplied to the left, while a trinomial is multiplied by a 4-term polynomial to the right.  Each arrow represents an individual multiplication that must be performed in the problem.  As the two polynomials being multiplied increase in size, the number of individual multiplications also increases.

Lesson

Multiplying_Polynomials_2_visual_5

In this lesson, you will learn how to multiply a binomial times another binomial.  This process involves taking each individual term of the first binomial and multiplying it by each term of the second binomial.

Suppose that you are multiplying the following binomials:  x + 2 and x + 3.  Since both terms are part of the binomial, use parenthesis for the multiplication.  A good way to write the problem is by simply writing (x + 2)(x + 3) without any kind of multiplication sign.  Most first time problem solvers assume that the problem is done like this:


Multiplying_Polynomials_2_visual_6

 

 

However, this answer can be shown to be wrong by picking a number for x and seeing whether the problem value is the same as the answer value.  Suppose that x = 3.  The problem value and answer value are shown below.

Multiplying_Polynomials_2_visual_7

 

 

The correct way to do this problem is to multiply each term in the first binomial times each term in the second binomial.

The method of multiplying a binomial times another binomial requires four separate multiplications.  These four separate multiplications can be represented by the acronym “foil”.  The problem below shows how each multiplication can be represented by a letter in the word “foil”.

Multiplying_Polynomials_2_visual_8

 

Look at the above example carefully.  See if you can get a good idea of what each letter in F-O-I-L stands for.  Here is another example:  Multiply (x + 5)(x – 2)


Multiplying_Polynomials_2_visual_9

 

The above problem contains a subtraction sign.  The mathematically correct way to do the problem is to consider that x – 2 really equals x + (-2).  Use the (-2) in multiplication to get the correct answer.


Multiplying_Polynomials_2_visual_10A good way to think of the distributive property is as a “crooked rainbow” where an individual part of a rainbow connects the initial (monomial) expression to each term of the second (polynomial) expression.  Each part of the rainbow represents a multiplication.

 

 

The crooked rainbow is a good way to show the foil method in one step.  This time there will be two separate rainbows, one coming from each term in the first binomial.  Put the second rainbow below the problem to make it easier to read.  See if you can do the work and answer for each of the following problems.


Multiplying_Polynomials_2_visual_11aMultiplying_Polynomials_2_visual_11bMultiplying_Polynomials_2_visual_11cMultiplying_Polynomials_2_visual_11e

 

 

 

Multiplying_Polynomials_2_visual_12aMultiplying_Polynomials_2_visual_12bMultiplying_Polynomials_2_visual_12cMultiplying_Polynomials_2_visual_12e

 

When the numbers get larger, follow the same procedure.  The following problem contains coefficients on the variables as well as larger numbers as the constants.

Multiplying_Polynomials_2_visual_13a

 

Multiplying_Polynomials_2_visual_13b

 


Here is one final reminder of the foil method.  Scroll over the problem below to see how each letter in foil is applied in the multiplication.

Multiplying_Polynomials_2_visual_14a

 

Multiplying_Polynomials_2_visual_15a

Try It

Multiply each pair of expressions:


1)  (a + 3)(a + 7)

2)  (2b + 5)(b – 5)

3)  2c (4c2 – 3c + 15)

4)  (3d + 5)(-2d – 4)

5)  (8e2 + 5)(e2 + 3)


Scroll down for answers

 

 

 

 

 

 

 

 

 


Answers:

1)  a2 + 10a + 21

2)  2b2 – 5b – 25

3)  8c3 – 6c2 + 30c

4)  -6d2 – 22d – 20

5)  8e4 + 29e2 + 15

 


 

 

Related Links:

Didn't find what you were looking for in this lesson?  More information on polynomials can be found at the following places:


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