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### What are Like Terms?

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Date: 11/20/2001 at 23:06:30
From: Carlos Mena
Subject: I really need help.

Hi,

I'm a student at Herbert Hoover Middle School and in math we are doing
a project. What are like terms?
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```
Date: 11/21/2001 at 11:15:57
From: Doctor Ian
Subject: Re: I really need help.

Hi Carlos,

A 'term' is the product of some constants (numbers whose values we
know) and variables (numbers whose values we don't know). For example,

3
3x
3x^2
3xy^2
3xy^2z^3

are all 'terms'.  (Another word for 'term' is 'monomial'. If you add a
bunch of monomials together, you get a polynomial.)

Two terms are 'like terms' (or 'alike' terms, which in many contexts
seems clearer) if their variable patterns are the same. Here is a test
you can do: arrange all the variables in alphabetical order, and see
if you have any mismatches:

3 x
5 x
-
|
+------------> These patterns match, so these are like terms.

3        x
2 * pi * x

-
|
+-----> These patterns match, so these are like terms.

3 x
3 y
-
|
+------------> These have different variables, so they are
NOT like terms.

3 x
3 x y
-
|
+----------> These have different variables, so they are
NOT like terms.

3 x^2 y^2
3 x   y^2
---
|
+-----------> These have the same variables, but different
exponents for x, so they are NOT like terms.

Note that the 'nomial' in monomial/polynomial means 'name', so the
'name' of a term is the pattern of variables that it contains.

Actually, the variables are kind of like last names, and the constants
- also called 'coefficients' - are like first names.  This isn't an
exact analogy: We would say that'John Smith' and 'Joe Smith' are
similar names, but we would also say that 'John Smith' and 'John
Jones' are similar. However, when talking about terms, we only pay
attention to the last names:

First
name
--
3 xy^2z
Same first names, same last names: 'equal'
3 xy^2z
-----
Last
name

First
name
--
3 xy^2z
Different first names, same last names: 'like'
21 xy^2z
-----
Last
name

First
name
--
3 xy^2z
Same first names, different last names: 'NOT like'
3 x^2yz
-----
Last
name

anything else.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Middle School Algebra