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 x’s and another seven x’s. Clearly, the end result is a total of ten x’s. 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.

Example:

Simplify: 5x ² + 9 – 3x + 4x ² + 8x + 7.

solution:

This expression has six terms altogether. However, we notice that

• 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 = 9x 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 combined arithmetically:

9 + 7 = 16.

So

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.