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Multiplying Binomial Expressions

by Ron Kurtus (3 April 2008)

A binomial is an expression consisting of two terms or monomials separated by a plus (+) or minus (−) sign. Examples of binomials include ax + b, x² − y² and 2x + 3y. Even x + 2 + 7 is a binomial, since it reduces to x + 9, which has two terms. Multiplying two binomial expressions can be similar to multiplication of numbers. Typically, the FOIL method is used to multiply binomials. Multiplying binomial expressions is often used to verify the factoring of a trinomial or quadratic expression.

Questions you may have are:

• How is multiplying binomials similar to multiplying numbers?
• What is the FOIL method?
• x?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Similar to multiplying numbers

Multiplying binomials is similar to multiplying two 2-digit numbers. Suppose you wanted to multiply 21 × 32. You could write the multiplication as:

    21
  ×32
____
    42
  63
____
672

Break into binomial form

Another perspective on it is break the numbers into a form similar to a binomial:

         20 + 1
        ×30 + 2
       _______
           40 + 2
600 + 30
___________
600 + 70 + 2 = 672

Note: I hope it lined up on your computer.

Multiply two binomials

Now, you can use the same method to multiply two binomials together:

          2x − 7
        × x + 3
         _______

           6x − 21
2x² − 7x
___________
2x² − x − 21

Breaking it down that multiplication, step-by-step is:

3 * 7 = 21

3 * 2x = 6x

x * −7 = −7x

x * 2x = 2x²

Note: To avoid confusion between x and the multiplication sign ×, we used the alternate * for times.

FOIL method

Since binomials are simple and you are smart, a FOIL method is usually used to multiply two binomials. FOIL stands for multiply the First terms, multiply the Outside terms, multiply the Inside terms and multiply the Last terms.

To multiply (ax + b)(cx + d), you follow the procedure:

1. Multiply the First terms together: (ax)(cx) = acx²

2. Multiply the Outside terms: (ax)(d) = adx

3. Multiply the Inside terms: (b)(cx) = bcx

4. Multiply the Last terms: (b)(d) = bd

5. Add the results to get: acx² + adx + bcx + bd

Typically, you can do these operations in your head, writing down the results in their order.

Example

Multiply (x + 3y)(2x − 5y)

1. Multiply the First terms: (x)(2x) = 2x²

Write down: 2x²

2. Multiply the Outside terms: (x)(−5y) = −5xy

Add − 5xy: 2x² − 5xy

3. Multiply the Inside terms: (3y)(2x) = 6xy

Add 6xy: 2x² − 5xy + 6xy = 2x² + xy

4. Multiply the Last terms: (3y)(−5y) = −15y²

Add −15y² for final result: 2x² + xy − 15y²

Special form

Another example is: (2x + 3y)(2x − 3y) =

4x²

4x² − 6xy

4x² − 6xy + 6xy = 4x²

4x² − 9y²

Note: When you multiply binomials in the form of (ax + by)(ax − by), the result is a²x² − b²y². Remember this shortcut, because it will come up time and time again in Algebra. (x − 2)(x + 2) = x² − 4

When binomials not similar

When the binomials are not similar, it can get tricky:

(x²− y)(x − 2y) =

(x³) + (−2yx² − xy) + 2y² =

x³ − 2yx² − xy + 2y²

Exercises

Try the following exercises:

1. (5x − 7)(x + 2)

2. (x² + 3)(x² + 3)

3. (x − y)(3x − 2y)

4. (4 − y)(y + 4)

5. (a + b)(c + d)

Answers

1. 5x² + 3x − 14

2. x⁴ + 6x² + 9

3. 3x² − 5y + 2y²

4. y² − 16

5. ac + ad + bc + bd

Summary

Multiplying two binomial expressions can be similar to multiplication of numbers. Typically, the FOIL (First, Outside, Inside, Last) method is used to multiply binomials. Multiplying binomial expressions is often used to verify the factoring of a trinomial or quadratic expression.

Answers to Readers' Questions

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Resources

The following resources provide information on this subject:

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Books

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Miscellaneous

Mini-quiz to check your understanding

1. Is 5 + 4x − 3 a binomial?

2. What is (x + 3)² equal to?

3. What is the product of (5x − 7)(x + 2)?

If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.

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