

Terms are the parts of an expression that are being added and/or subtracted. 


This expression, 2x + 5, has two terms (2x and 5).
One term (2x) contains a variable, x.
The other term (5) is a number, referred to as a constant.


Like terms (or similar terms) are terms that represent the same quantities.
In algebra, these terms have the exact same variable part.


Consider the statement: Two cats plus three more cats equals a total of five cats.
If we represent "cats" by the letter (variable) "c", we have, 2c + 3c = 5c.
In all three terms, "c" represents the same thing (the species "cat"),
so, the "c's" are referred to as "like" terms.
Since the "c's" are like terms, their quantities can be added together.
Terms that are not "like terms" cannot be added together.
Two cats and three dogs cannot be added together.
2c + 3d = 2c + 3d
Cats (c) and dogs (d) are not "like" terms, and cannot be added.
Like terms, are terms whose variables (including any exponents) are the same.
They are terms that look like each other.
The only difference will be the coefficients (the numbers in front of the variables).
Like Terms:

Notes: 
3, 12, 4, 5x^{0} 
The simplest "like terms" are numbers (constants).
Remember that the "variable" in a constant term has an exponent of 0, making the variable equal to 1. 
3x^{2 }, 6x^{2 }, 2x^{2 }, x^{2 } 
These are "like terms" because each has the variable factor of x^{2 }. 
4a^{2}b^{3}, a^{2}b^{3}, 3a^{2}b^{3} 
These are "like terms" because each has the variable factor of a^{2}b^{3}. 
Not Like Terms: 
Notes: 
3, 4xy, 2a, 5x^{2}y 
These terms do not have the same variable factor. Remember that xy and x^{2}y are different terms. These examples are unlike terms. 
Like terms can be combined under addition or subtraction.
This process is the same as adding apples to apples, and oranges to oranges.
Just remember to add only the terms that are "like".
Problem: 
Answer: 
• Simplify: 2x + 4x + x 
Don't forget the implied "1" in front of x.
Think of it as two x's plus four more x's plus one more x.
You can also factor: x(2 + 4 + 1) = 7x
Answer: 7x 
• Simplify: 2x^{2} + x  5 + x^{2 }  4x + 9 
Group the like terms together and them combine them. Remember when moving terms to move their signs with them. 2x^{2} + x^{2} + x  4x  5 + 9
Factor: x^{2} (2 + 1) + x (1  4)  5 + 9
Answer: 3x^{2}  3x + 4

• Simplify: 6x^{2}  (10 + 3x^{2 }  2x) 
Deal with parentheses before you combine the like terms. In this case, distribute 1 across the parentheses. 6x^{2} 1(10 + 3x^{2 }  2x)
6x^{2} 10  3x^{2 }+ 2x = 6x^{2}  3x^{2 }+ 2x 10
Answer:
3x^{2} + 2x 10

• Simplify: a^{2}b + 2(3a^{2}b  5a) 
Distribute the +2 across the parentheses.
a^{2}b + 2(3a^{2}b  5a) = a^{2}b + 6a^{2}b  10a
Answer: 7a^{2}b  10a

• Simplify: 5x^{2}  4x^{3}  x^{2 }+ 8 
Look carefully at the exponents.
Answer:  4x^{3} + 4 x^{2 }+ 8 
Additional vocabulary associated with "terms":
If any, and all, exponents appearing in a term are whole numbers (0, 1, 2, 3, ...),
a term may be referred to as a monomial.
Classification by Terms 
monomial 
one term: 12, 4x, x^{2}, 5xy 
binomial 
two terms: 2x  1, x^{2 } 4 
trinomial 
three terms: x^{2} + 2x + 1 
polynomial  one or more terms
polynomial means "many",
but it can also be one term. 
The ending of these words "nomial" is Greek for "part". 
Classification by Degree 
Linear  degree of 1 or 0: 3x + 1 or 12 
Quadratic  degree of 2: 2x^{2}  x + 7 
Cubic  degree of 3: 3x^{3 } + 4x^{2} + 3x + 5 
The degree of a term with whole number exponents is the sum of the exponents of the variables, if there are variables. Nonzero constants have degree 0, and the term zero has no degree.
Example: 6x^{2} has a degree of 2
4x^{2}y^{3} has a degree of 5 (the sum of 2 and 3) 
The degree of a polynomial is the highest degree of its terms.
Example: 3x^{2} + 4x + 1 has a degree of 2
x^{3}  x^{2} + 5x  2 has a degree of 3 
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