What are Like Terms?

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?

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

 
Does this help?  Write back if you'd like to talk more about this or 
anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/