IDENTIFYING VARIABLE PARTS AND COEFFICIENTS OF TERMS
by Dr. Carol JVF Burns (website creator)
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• PRACTICE (online exercises and printable worksheets)
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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 nameit 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: $\,{\bf -6xy}\,$
Master the ideas from this section

When you're done practicing, move on to:
Combining Like Terms

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

Consider the term:
What is the numerical coefficient?

What is the variable part?

Write the term in the most conventional way: