Multiplying Binomial Expressions
by Ron Kurtus (revised 18 January 2022)
A binomial expression is an algebraic 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.
In Algebra, you are often required to multiply expressions together. The best way to multiply two binomial expressions is to use what is called the FOIL method. In this method, you multiply the First, Outside, Inside, and Last terms and then add them together.
You should follow some good practices, such as putting terms in the right order, before you start. Also, there are a few shortcuts that are good to remember.
Questions you may have include:
- What is the FOIL method?
- What are some good practices?
- What are some special situations?
This lesson will answer those questions.
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." Then you add the results together in the proper order.
For example, to multiply (ax + b)(cx + d), you follow the procedure:
1. Multiply the two First terms together (ax + b)(cx + d): (ax)(cx) = acx2
2. Multiply the two Outside terms (ax + b)(cx + d): (ax)(d) = adx
3. Multiply the two Inside terms (ax + b)(cx + d): (b)(cx) = bcx
4. Multiply the two Last terms (ax + b)(cx + d): (b)(d) = bd
5. Add the results to get: acx2 + adx + bcx + bd
This can also be written as: acx2 + (ad + bc)x + bd
Typically, you can do these operations in your head, writing down the results in their order.
There are different situations you can study.
It is a good practice to arrange the terms in a uniform order. For example, supposed you want to multiply (3y + x)(2x − 5y). Although you will get the same answer using that order, it is best and easier to rearrange the terms as (x + 3y)(2x − 5y).
Then multiply (x + 3y)(2x − 5y):
1. First terms: (x)(2x) = 2x2
2.Outside terms: (x)(−5y) = −5xy
3. Inside terms: (3y)(2x) = 6xy
4. Last terms: (3y)(−5y) = −15y2
5. Add together: 2x2 −5xy + 6xy −15y2
6. Combine like terms to get the final result: 2x2 + xy − 15y2
Likewise, you should rearrange (xa − 7)(5x + 2) to be (ax − 7)(5x + 2) before multiplying.
You also want to simplify expressions, if possible. For example, the expression 2x + 3 − 8 can be simplified into a binomial expression: 2x −5.
There are special situations to be aware of.
When binomials not similar
When the binomials are not similar, it can get tricky. For example, multiply (x2 − y)(x − 2y):
1. First: (x2)(x) = x3
2.Outside: (x2)(−2y) = −2x2y
3. Inside: (−y)(x) = −xy (note that we changed the order of x and y)
4. Last: (−y)(−2y) = 2y2
5. Add together: x3 − 2x2y − xy + 2y2
This result has four terms instead of the usual three terms.
Squaring an expression
If you are going to multiply an expression by itself, such as (ax + b)2 = (ax + b)(ax + b), it is easy to get the result without the FOIL method.
(ax + b)(ax + b) = a2x2 + 2abx + b2
(ax − b)(ax − b) = a2x2 − 2abx + b2
It is good to remember this shortcut.
When you multiply binomials in the form of (a + b)(a −b), the result is a2 − b2.
For example, multiply (2x + 3y)(2x − 3y)
1. First: (2x)(2x) = 4x2
2. Outside: (2x)(−3y) = −6xy
3. Inside: (3y)(2x) = 6xy
4. Last: (3y)(−3y) = −9y2
5. Add together: 4x2 −6xy + 6xy −9y2 for the final result:
4x2− 9y2 or 22x2− 32y2
Remember this shortcut, because it will come up time and time again in Algebra.
Another example is simply: (x − 2)(x + 2) = x2 − 4
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)
1. 5x2 + 3x − 14
2. x4 + 6x2 + 9
3. 3x2 − 5xy + 2y2
4. 16 − y2
5. ac + ad + bc + bd
You are often required to multiply expressions together. The best way to multiply two binomial expressions is to use what is called the FOIL method.
There are some good practices to follow, such as putting terms in the right order.
Some shortcuts to remember are:
(a + b)(a + b) = a2 + 2ab + b2
(a − b)(a − b) = a2 − 2ab + b2
(a + b)(a − b) = a2 − b2
Use clever tools to make work easier
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Multiplying Binomial Expressions