Collecing Like Terms
Like terms in an algebraic expression are
terms with identical symbolic or variable parts. Thus
3x and 7x are like terms because both
contain the symbolic part x
5x 2 yz and 13x 2
yz are like terms because both contain the symbolic part
x 2 yz
4x 2 and 7x are not
like terms because even though the symbol present in both is
x, the symbolic part x 2 is
not identical to the symbolic part x.
Algebraic expressions that contain like terms can be
simplified by combining each group of like terms into a single
term. The reason why this is possible and valid is quite easy to
see. For instance, consider the expression
3x + 7x
which is the sum of two like terms, representing the
accumulation of three xs and another seven xs.
Clearly, the end result is a total of ten xs. In notation
3x + 7x = (3 + 7)x = 10x
This process of combining (or collecting )
like terms can be performed for each group of like terms that
appear in an expression. The net effect will be that the original
expression can now be written with fewer terms, yet which are
entirely equivalent to the terms in the original expression.
Simplify: 5x 2 + 9 3x + 4x 2 + 8x
This expression has six terms altogether. However, we notice
- two of the terms have the literal part x 2
and so are like terms we can replace
5x 2 + 4x 2 by (5 + 4)x 2 =
two of the terms have the same literal part x and
so are also like terms. We can replace
-3x + 8x by (-3 + 8)x = 5x
two of the terms are just constants, and so obviously can be
9 + 7 = 16.
5x 2 + 9 3x + 4x 2 + 8x + 7
= 5x 2 + 4x 2 + (-3x) + 8x + 9 + 7
= (5 + 4)x 2 + (-3 + 8)x + 16
= 9x 2 + 5x + 16.
Thus, in simplest form, the original six term expression can
be rewritten as
9x 2 + 5x + 16
consisting of just three terms.