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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 include:
- How is multiplying binomials similar to multiplying numbers?
- What is the FOIL method?
- What are some exercises on using the FOIL method?
This lesson will answer those questions.
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 = 672Note: 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
2. Multiply the Outside terms: (x)(−5y) = −5xy
3. Multiply the Inside terms: (3y)(2x) = 6xy
4. Multiply the Last terms: (3y)(−5y) = −15y2
5. Add together: 2x2 −5xy + 6xy −15y2 for final result:
2x2 + xy − 15y2
Special form
Another example is: (2x + 3y)(2x − 3y) =
First: 4x2
Outside: − 6xy
Inside: 6xy
Last: − 9y2
Add together: 4x2 − 6xy + 6xy − 9y2 = 4x2− 9y2
But also 4x2− 9y2 = 22x2− 32y2
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 − 5xy + 2y2
4. 16 − y2
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.
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