Algebra Homework Help : Simplification - Combining Like Terms

How to Find the GCF in Two or More Algebraic Terms ?

Like terms are those that have exactly the same variables and exponents. They may differ only in their coefficients.

Examples of Like Terms :
x, 2x, 25x, -3x, and -48x (have the same variable x¹).
1, 2009, -32, -452, and -3.07 (are constants having the same variable x⁰).
a², 4a², -a², -21a², and 0.26a² (have the same variable a²).
7y²z and -9y²z (have the same variables y²z).

Examples of Unlike Terms :
23x and 23y (have different letter variables x and y).
z, z² and -z³ (have the same letter variable, but raised to a different exponent).
0.1xyz and -0.1xy (both have the variables xy, but not z). x²y and xy² (both have the variables xy, but raised to different exponents).

Importantly, the only algebraic terms that can be combined (added or subtracted) are like terms. Since like terms differ only in their numerical coefficients, they are simply combined by adding and subtracting their coefficients. Combining like terms therefore implies adding or subtracting like terms and obtaining simplified equivalent expressions.

Example : 8x⁴ - 17x + 9 + 3x⁴ +12x + 27 = 11x⁴ - 5x + 36.
Here, 8x⁴ + 3x⁴ = 11x⁴ (combining like terms in x⁴).
Next, -17x +12x = -5x (combining like terms in x).
Finally, 9 + 27 = 36 (combining the constants).

Note that a variable having an exponent of one is written without an exponent (i.e., x¹ = x) and a variable with an exponent of zero equals one (i.e., x⁰ = 1).

Practice Exercise for Algebra Module on Combining Like Terms