# Identifying Variable Parts and Coefficients of Terms

## 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: $\,{\bf {-}6}\,$

What is the variable part?  Answer: $\,{\bf xy}\,$

Write the term in the most conventional way:  Answer: $\,{\bf -6xy}\,$

## Practice

When reporting the variable part, variables must be written in alphabetical order.

What is the numerical coefficient?
What is the variable part?
Write the term in the most conventional way: