Like terms

In algebra, like terms are terms that have the same variables and powers. The coefficients do not need to match.

Unlike terms are two or more terms that are not like terms, i.e. they do not have the same variables or powers. The order of the variables does not matter unless there is a power. For example, 8xyz² and −5xyz² are like terms because they have the same variables and power while 3abc and 3ghi are unlike terms because they have different variables. Since the coefficient doesn't affect likeness, all constant terms are like terms.

Generalization

In this discussion, a "term" will refer to a string of numbers being multiplied or divided (remember that division is simply multiplication by a reciprocal) together. Terms are within the same expression and are combined by either addition or subtraction. For example, take the expression:

$ax+bx$

There are two terms in this expression. Notice that the two terms have a common factor, that is, both terms have an $x$. This means that we can factor out that common factor variable, resulting in

$(a+b)x$

If the expression in parentheses may be calculated, that is, if the variables in the expression in the parentheses are known numbers, then it is simpler to write the calculation $a+b$. and juxtapose that new number with the remaining unknown number. Terms combined in an expression with a common, unknown factor (or multiple unknown factors) are called like terms.

Examples

Example

To provide an example for above, let $a$ and $b$ have numerical values, so that their sum may be calculated. For ease of calculation, let $a=5$ and $b=3$. The original expression becomes

$5x+3x$

which may be factored into

$(5+3)x$

or, equally,

$8x$.

This demonstrates that

$5x+3x=8x$

The known values assigned to the unlike part of two or more terms are called coefficients. As this example shows, when like terms exist in an expression, they may be combined by adding or subtracting (whatever the expression indicates) the coefficients, and maintaining the common factor of both terms. Such combination is called combining like terms, and it is an important tool used for solving equations.

Simplifying an expression

Take the expression, which is to be simplified:

$3(4x^{2}y-6y)+7x^{2}y-3y^{2}+2(8y-4y^{2}-4x^{2}y)$

The first step to grouping like terms in this expression is to get rid of the parentheses. Do this by distributing (multiplying) each number in front of a set of parentheses to each term in that set of parentheses:

$12x^{2}y-18y+7x^{2}y-3y^{2}+16y-8y^{2}-8x^{2}y$

The like terms in this expression are the terms that can be grouped together by having exactly the same set of unknown factors. Here, the sets of unknown factors are $x^{2}y,$ $y^{2},$ and $y.$. By the rule in the first example, all terms with the same set of unknown factors, that is, all like terms, may be combined by adding or subtracting their coefficients, while maintaining the unknown factors. Thus, the expression becomes

$11x^{2}y-2y-11y^{2}$

The expression is considered simplified when all like terms have been combined, and all terms present are unlike. In this case, all terms now have different unknown factors, and are thus unlike, and so the expression is completely simplified.