Binomials and FOIL Method

What is a binomial?

A binomial is an algebraic expression of the sum (+) or the difference (-) of two terms.

 

Quick Review:

Let’s review some terms and expressions that might help us understand these.

* = multiplication

/ = division

Monomial examples: 5y, -8x2, 3

Binomial examples: -3x2 *– 2, 9y – 2y2

Polynomial examples: 8x2 + 3x -2, 15x2 + 18x – 35x - 42

 

What is a polynomial?

Polynomials are algebraic expressions that include real numbers (positive, negative, large, small, whole, or decimal numbers) and variables (x, y, etc.).  They include more than term and are the sum of monomials.  They are usually also written in decreasing order of terms.

 

Term Polynimal or Not? Why?
8x2 + 3x -2 Polynomial  
8x-3 - 7y -2 NOT a polynomial The exponent is negative (-3)
8x+ 8x -2/3 NOT a polynomial Cannot have division 

 

Now...how do we multiply binomials?

When multiplying binomials, you can use the FOIL method. For instance, to find the product of 2 binomials, you’ll add the products of the First terms, the Outer terms, the Inner terms, and the Last terms.

 

FIRST: multiply the first term in each set of parenthesis

OUTER: multiply the outer term in each set of parenthesis

INNER: multiply the inner term in each set of parenthesis

LAST: multiply the last term in each set of parenthesis

 

Example 1

Let’s work out this problem.

(3x – 7) * (5x + 6)

 

1. Identify the FOIL numbers

FIRST: (3x) * (5x)

OUTER: (3x) * (6)

INNER: (-7) * (5x)

LAST: (-7) * (6)

 

2. Multiply the terms

FIRST: (3x) * (5x) = 15x2

OUTER: (3x) * (6) = 18x

INNER: (-7) * (5x) = –35x

LAST: (-7) * (6) = –42

 

3. Combine like terms

18x  –35x = –17x

 

4. Combine all terms in decreasing order

15x2 –17x –42

 

Final Answer: 15x2  –17x –42

 

 

Example 2

(5 + 4x) * (3 + 2x)

 

 

(5 + 4x) * (3 + 2x) = (5 + 4x) (3 + 2x)

= (5)(3) + (5)(2x) + (4x)(3) + (4x)(2x)

= 15 + 10x + 12x + 8x2

= 15 + 22x + 8x2

8x2 + 22x + 15

 

Final answer: 8x2 + 22x + 15

Additional Resources

Our Math Tutors recommend the following websites for help:

Paul's Online Notes: Polynomials