CB-MA007-261013: Identifying Variable Parts and Coefficients of Terms

Read these notes to get a handle on the definitions we'll be using and to see a few examples. After you read, complete a few practice problems and check your answers.

Numerical Coefficients and Variable Parts

In a term like $$2x$$, there are two parts that are usually of interest: the numerical part (2) and the variable part $$(x)$$.

The numerical part is given a special name - it is called the numerical coefficient or, more simply, the coefficient of the term.

In the term $$4xy$$, the coefficient is 4 and the variable part is $$xy$$.

In the term $$−7x^2y^3$$, the coefficient is −7 and the variable part is $$x^2y^3$$.

Coefficients 1 and −1

If you don't 'see' a coefficient, then it is 1. That is, $$x=(1)x$$ has coefficient 1. Also, $$x^2y^3=(1)x^2y^3$$ has coefficient 1. It's never necessary to write a coefficient of 1, because multiplication by 1 doesn't change anything.

In the term $$−x$$, the coefficient is −1, because $$−x=(−1)x$$. In the term $$−x^2y$$, the coefficient is −1, because $$−x^2y=(−1)x^2y$$.

Constant Terms

A term like 2 that has no variable part is called a constant term, because it is constant - it never changes. It has no variable part that can 'hold' different values. Thus $$\frac{1}{2}$$, $$\sqrt 3$$, and 9.4 are all constant terms.

Conventional Way to Write Terms

In any term, it is conventional to write the numerical coefficient first. Thus, you should write $$4xy$$, not $$xy4$$ or $$x4y$$.

Also, it is conventional to write any variable(s) in alphabetical order. Thus, you usually want to write 5xy, not $$5yx$$. Similarly, write $$x^2yz$$, not $$yx^2z$$ or $$yzx^2$$.

Example

For this web exercise, when reporting the variable part, variables must be written in alphabetical order. The user input is in bold in the example below.

Consider the term $$y(3)(2x)(−1)$$.

What is the numerical coefficient? Answer: −6

What is the variable part? Answer: $$xy$$

Write the term in the most conventional way:

Answer: $$−6xy$$

Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/id_var_part_coeff.htm
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