FOIL & The Distributive Property (3 Things To Know)


FOIL and the Distributive Property are both used often in mathematics, starting in Algebra.  However, one stems from the other, and it is important to know when to use each one.

So, what is the difference between FOIL & the Distributive Property?  FOIL is only used for a product of two binomials, while the Distributive Property is used for any product of polynomials.  FOIL is an acronym that stands for First, Outer, Inner, Last.  It helps us to avoid forgetting or repeating any product of terms from the two binomials.

Of course, we can extend the basic form of the Distributive Property to any number of summed terms in parentheses, which allows us to use it for any product of polynomials.

In this article, we’ll take a closer look at FOIL and the Distributive Property, including when to use each one and how FOIL comes from the Distributive Property.

Let’s get started.

What Is The Distributive Property?

The Distributive Property is a way to multiply a term across a sum of two terms in parentheses.  In its simplest form, the equation for the Distributive Property is given by:

  • A(B + C) = AB + AC  [simplest form of Distributive Property]

For example, by the Distributive Property:

  • 2*(3 + 4) = 2*3 + 2*4  [by Distributive Property]
  • =6 + 8
  • =14

We can verify this solution by working inside parentheses first and then multiplying:

  • 2*(3 + 4) = 2*(7)
  • = 14

Of course, we can also extend the Distributive Property to a sum of three or more terms:

  • A(B + C + D) = AB + AC + AD  [Distributive Property for a sum of 3 terms]
  • A(B + C + D + E) = AB + AC + AD + AE  [Distributive Property for a sum of 4 terms]

and so forth.  The Distributive Property works for any terms, including those with variables.

For example, 2(x + y + z) = 2x + 2y + 2z.

When To Use The Distributive Property

The Distributive Property is most often used to simplify parentheses (for example, to multiply a sum of two or more terms in parentheses by another term).

However, it can be used for any product of polynomials, including:

  • A monomial times a binomial (the simplest form of the Distributive Property).
  • A binomial times a binomial (there is a shortcut for this method, called FOIL, which we will discuss later).
  • A binomial times a trinomial (this would involve repeated use of the Distributive Property).

Examples Of The Distributive Property

Here are some examples of how to use the Distributive Property.

Example 1: Use The Distributive Property To Solve An Equation

In this example, we will use the Distributive Property to simplify an expression and solve an equation.

Let’s say we want to solve the equation 3(2x + 5) = 27.

First, we use the Distributive Property on the sum of two terms in parentheses on the left side of the equation:

  • 3(2x + 5) = 27  [original equation]
  • 3*2x + 3*5 = 27  [by the Distributive Property]
  • 6x + 15 = 27  [simplify products on left side of equation]
  • 6x = 12  [subtract 15 from both sides of equation]
  • x = 2  [divide by 2]

Thus, the solution is x = 2.  Note that we can also solve this equation if we take these steps: divide both sides by 3, subtract 5 from both sides, and then divide by 2.  (Think about why this works!)

Example 2: Use The Distributive Property To Multiply Two Polynomials

Let’s say we want to multiply and simplify (x + y)*(a + b + c).

First, we’ll replace x + y to make things simpler.  Let’s say z = x + y.  Then:

  • (x + y)*(a + b + c)  [original expression]
  • =z*(a + b + c)  [let z = x + y]
  • =za + zb + zc  [Distributive Property for a sum of three terms in parentheses]
  • = (x + y)a + (x + y)b + (x + y)c  [since z = x + y]
  • = xa + ya + xb + yb + xc + yc  [used the Distributive Property three times]

What Is The FOIL Method?

The FOIL method is used to multiply the product of two binomials.  The FOIL acronym stands for “first, outer, inner, last, or:

  • F: first – take the product of the first term in each parentheses (left term from each parentheses)
  • O: outer – take the product of the outer term in each parentheses (left term from first parentheses, right term from second parentheses)
  • I: inner – take the product of the inner term in each parentheses (right term from first parentheses, left term from second parentheses)
  • L: last – take the product of the last term in each parentheses (right term from each parentheses)

The FOIL acronym helps us to avoid forgetting or repeating any product of terms from the binomials.  The formula is given by:

  • (A + B)(C + D) = AB + AD + BC + BD  [Formula for FOIL Method]

The picture below gives a visual that might help to explain where the terms come from:

FOIL Method Graphic
The FOIL acronym is a shortcut for repeated use of the Distributive Property. It is used in the special case of a product of two binomials, and it helps us to avoid forgetting or repeating any product of terms from the binomials.

For example:

  • (2 + 3)(4 + 5)
  • =2*4 + 2*5 + 3*4 + 3*5  [used FOIL formula]
  • 8 + 10 + 12 + 15
  • =45  [same answer as 5*9]

Since any binomial has two terms, we will always get four terms from the product of two binomials.  However, it may be possible that there are like terms to be combined.

For example, let’s say we take the product (x + 1)(x + 2).  FOIL gives us:

  • (x + 1)(x + 2)
  • =x*x + x*2 + 1*x + 1*2
  • =x2 + 2x + x + 2
  • =x2 + 3x + 2  [the terms 2x and x are like terms, so we can add them to get 3x]

Remember that the FOIL Method is can only be used for a product of two binomials.  For any other product of polynomials, we must use the Distributive Property.

Also remember that the FOIL Method is a shortcut for repeated use of the Distributive Property, which we will prove now:

  • (A + B)(C + D)  [original product of two binomials]
  • =E(C + D)  [let E = A + B]
  • = EC + ED  [used the Distributive Property]
  • = (A + B)C + (A + B)D  [since E = A + B]
  • = AC + BC + AD + BD  [used the Distributive Property two times]
  • = AC + AD + BC + BD  [rearranged terms]

The last line is the same formula that we get from the FOIL Method.

When To Use The FOIL Method

As mentioned earlier, the FOIL Method can only be used for a product of two binomials.  For any other product of polynomials, we need to use the Distributive Property.

One special case when we would use the FOIL method is to square a binomial.

Examples Of The FOIL Method

Here are some examples of how to use the FOIL Method.

Example 1: Using The FOIL Method To Square A Binomial

Let’s say we want to square the binomial 2x + 3.

Then we want to calculate:

  • (2x + 3)2
  • =(2x + 3)(2x + 3)  [a binomial squared is the binomial times itself]
  • =2x*2x + 2x*3 + 3*2x + 3*3  [used the FOIL Method]
  • =4x2 + 6x + 6x + 9  [simplified products of terms]
  • =4x2 + 12x + 9  [combined like terms]

There is a shortcut formula for the square of a binomial:

  • (A + B)2 = A2 + 2AB + B2  [formula for square of a binomial]
Example 2: Using The FOIL Method To Verify The Difference Of Squares Formula

We can also use FOIL to verify the formula for a difference of squares.  Remember that the formula to factor a difference of squares is given by:

  • A2 – B2 = (A + B)(A – B)  [equation to factor a difference of squares]

We can use FOIL on the right side of the equation to verify the formula:

  • (A + B)(A – B)
  • =A*A + A*-B + B*A + B*-B  [used FOIL Method]
  • =A2 – AB + BA – B2  [multiplied terms]
  • =A2 – AB + AB – B2  [BA = AB, by commutativity of multiplication]
  • =A2 – B2  [combined like terms: -AB + AB = 0]

What Is The Difference Between FOIL & The Distributive Property?

The main difference between FOIL and the Distributive Property is when you use each one:

  • FOIL can only be used in the special case when we have a product of two binomials.
  • The Distributive Property can be used for any product of polynomials.

Conclusion

Now you know a little more about FOIL, the Distributive Property, and how they are related.  You also know when to use FOIL (for a product of two binomials) and when to use the Distributive Property.

You can find a helpful video on multiplying binomials with the FOIL method on this page from Problem Solved Math.

For more on FOIL, check out this resource from Mesa Community College.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon

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