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