SIMPLIFYING ALGEBRAIC EXPRESSIONS/POLYNOMIALS
Like terms in an expression are those that have the same variables with the same exponents.
The coefficients may differ.
Example: 3m - 4n + 3 + 3m2 + 9mn - 21 + 3n2 - 2n + 12m - 5m3
3m and 12m are like terms
-4n and -2n are like terms
3 and -21 are like terms
Collect/combine like terms indicates add/subtract like terms in the expression. This is also true
for the directions given as "simplify" and "perform the indicated operations".
Example:
Simplify 3m - 4n + 3 - 5mn + 7n - 6 + 21m
(3m + 21m) + (-4n + 7n) + (3 - 6) - 5mn
(grouping like terms)
24m + 3n - 3 - 5mn
(add/subtract)
When an expression has grouping symbols, you need to simplify them following the order of
operations.
Example:
Simplify 4(3m - 4n + 3) - (3m2 + 3n2) - 2(5m - 13m2 + 8n2)
12m - 16n + 12 - 3m2 - 3n2 - 10m + 26m2 - 16n2
(There are no like terms inside
the grouping symbols to combine. Therefore, the next step is to use the distribute property so
that the grouping symbols are simplified.)
(12m - 10m) + (-3m2+ 26m2) + (-3n2 - 16n2) - 16n + 12 (group like terms)
2m + 23m2 - 19n2 - 16n + 12
(add/subtract)
NOTE: You do not need to write down the group like terms step. You can do that step along
with the add/subtract mentally.
EXAMPLES:
1) Simplify 4a - 5.7b + 3.3c - 12.4a + 2.2b - 5c
(4a - 12.4a) + (-5.7b + 2.2b) + (3.3c - 5c)
-8.4a - 3.5b - 1.7c
group like terms
add/subtract
NOTE: The final answer may be written in different order. However, since we have the
commutative property of addition, the solutions are equivalent.
For example, -8.4a - 3.5b - 1.7c is the same as -1.7c - 8.4a - 3.5b
2) Simplify 1/2x - 3(1/3 + 6x + 2y) + (1/2x + 6y - 1)
1/2x - 1 - 18x - 6y + 1/2x + 6y - 1
(1/2x - 18x + 1/2x) + (-6y + 6y) + (-1 - 1)
-17x - 2
distribute
group like terms
add/subtract
3) Simplify
2[x - (3 - x)]
2[x - 3 + x]
2[2x - 3]
4x - 6
remove parenthesis by distributing
combine like terms within parenthesis
remove bracket by distributing
NOTE: There may be more than one way to reach the same solution as long as you are correctly
following operations.
For example: 2[x - (3 - x)]
2[x - 3 + x]
remove parenthesis by distributing
2x - 6 + 2x
remove bracket by distributing
4x - 6
add/subtract
4) Simplify
9(x - 5) - 3(8 + 4x) + 3[x - (4 + x)]
9(x - 5) - 3(8 + 4x) + 3[x + 8 - x]
9x - 45 -24 -12x + 3x + 24 - 3x
-3x -45
given expression
distribute within brackets
remove parenthesis by distributing
add/subtract
5) Simplify
4[3(x2 + 5xy + 3y2) - (2x2 + 4y2)] - [12xy - 2(6x2 - xy - 3y2)]
given expression
4[3x2 + 15xy + 9y2 - 2x2 - 4y2] - [12xy -12x2 + 2xy + 6y2]
remove parenthesis by distributing
4[x2 + 15xy + 5y2] - [14xy - 12x2 + 6y2 ]
combine like terms within brackets
12x2 + 60xy + 20y2 - 14xy + 12x2 - 6y2
remove brackets by distributing
(12x2 + 12x2) + (60xy - 14xy) + (20y2 - 6y2)
group like terms
24x2 + 46xy + 14y2
add/subtract
6) Perform the indicated operations.
15 - 11{2- 5[3a + 4b + 2(4a - 3b + 6)]}
15 - 11{2 - 5[3a + 4b + 8a - 6b + 12]}
15 - 11{2 - 5[11a - 2b + 12]}
15 - 11{2 - 55a + 10b - 60}
15 - 11{-55a + 10b - 58}
15 + 605a - 110b + 638
653 + 605a - 11b
given expression
remove parenthesis by distributing
combine like terms within brackets
remove brackets by distributing
combine like terms within braces
remove braces by distributing
add/subtract
7) Simplify.
21 - {8 - 5x[6 - 4(y - 7) - 9(4 - y)] - 10x + 25xy + 13}
21 - {8 - 5x[6 - 4y + 28 - 36 + 9y] - 10x + 25xy + 13}
21 - {8 - 5x[-2 + 5y] - 10x + 25xy + 13}
21 - {8 + 10x - 25xy - 10x + 25xy + 13}
21 - {21}
0
given expression
remove parenthesis by distributing
combine like terms within brackets
remove brackets by distributing
combine like terms within braces
add/subtract