Terms, factors and co-efficients

Alge­bra is arith­metic with some of the num­bers not yet known. Before you can add, mul­ti­ply or sim­plify alge­braic expres­sions, you need names for their parts, just as you need the words "digit" and "place" before you can talk about dec­i­mals. This les­son gives you three such names: term, fac­tor and co-effi­cient. You will use them in every alge­bra chap­ter that fol­lows, from sim­pli­fy­ing expres­sions to solv­ing equa­tions.

A let­ter can stand for a num­ber

Think of a box of sweets. The lid is shut. You can­not count them. You still want to talk about that num­ber. So give it a let­ter. Call it $n$.

Now $n$ stands for the sweets in the box. The let­ter is not a code. It is not short for a word. It holds the place of a num­ber. You do not know that num­ber yet. Open the box and count $12$. Then $n$ is $12$.

Here is one more. You pick up a bag of rice. No one has weighed it. Call its weight $w$. You can talk about $w$ now. You do not need the scales first. Later the scales may say $3$. Then $w$ is $3$.

A let­ter like this is called a vari­able. It can stand for one num­ber today and a dif­fer­ent one tomor­row. Any let­ter will do. Peo­ple often use $x$, $y$ and $a$ because they are not short for any­thing. The let­ter is only a place for a num­ber to sit.

Now look at the box itself. It has one lid. That is $1$, and it stays $1$ what­ever is inside. A num­ber that never changes is called a con­stant. So $5$ and $-3$ are con­stants. This page is about what you build from these two parts.

Remem­ber. A let­ter stands for a num­ber. It waits for that num­ber to be filled in.

What a term is

A term is one sin­gle amount. It can be a con­stant on its own. So $6$ is a term. It can be a let­ter on its own. So $x$ is a term. It can be both, joined by mul­ti­ply­ing. So $6x$ is a term. It means $6 \times x$. Divid­ing counts as well.

Why do $\times$ and $\div$ stay inside a term? They build one amount out of the parts. What­ever $x$ is, $6x$ is one num­ber. Divid­ing does the same job. What­ever $x$ is, $\displaystyle \frac{x}{2}$ is still one num­ber.

Plus and minus do a dif­fer­ent job. They join one amount to another. So they show you where a term ends. In $3 + 2x$ there are two terms. They are $3$ and $2x$.

Remem­ber. Mul­ti­ply­ing and divid­ing stay inside a term. Plus and minus split one term from the next.

Worked exam­ple: terms in a longer expres­sion

List the terms of $5xy - 6x + 3$.

Read from left to right and cut at every $+$ or $-$. Each minus stays with the term after it.

$$5xy \quad\big|\quad -6x \quad\big|\quad +3$$

So there are three terms: $5xy$, $-6x$ and $3$. The first two con­tain vari­ables. The last is a con­stant term.

Worked exam­ple: a term with divi­sion

How many terms are in $\displaystyle 4a^2b - \dfrac{ab}{3} + 7$?

The divi­sion line in $\displaystyle \dfrac{ab}{3}$ stays inside its term, because it builds one amount. Cut only at the $-$ and the $+$. The terms are $4a^2b$, $\displaystyle -\dfrac{ab}{3}$ and $7$. That makes three.

Count the terms

How many terms are in each one? Count the parts split by $+$ and $-$.

Fac­tors

Mul­ti­ply things together and you get a prod­uct. So $5xy$ is a prod­uct. It means $5 \times x \times y$. Each part that goes in is a fac­tor. So $5$, $x$ and $y$ are the fac­tors of $5xy$.

You do this with plain num­bers too. $3 \times 5 = 15$. So $3$ and $5$ are fac­tors of $15$. Let­ters work the same way. A fac­tor can be a num­ber or a let­ter. Both of them count.

A fac­tor can also carry a minus sign. The minus stays with its own num­ber. So in $-4m$ the fac­tors are $-4$ and $m$.

Split­ting the prod­uct into its fac­tors

$$\begin{aligned}&5xy \\ &= 5 \times x \times y\end{aligned}$$

Remem­ber. A fac­tor is a part that is mul­ti­plied in.

Find the fac­tors

Write down the fac­tors of each prod­uct.

Co-effi­cients

Start with $5x$. It means $5 \times x$. That is five lots of $x$. The $5$ tells you how many. The $x$ tells you what you are count­ing.

Now give those two parts a name. The fac­tors of $5x$ are $5$ and $x$. Cover the $5$ with your fin­ger. What is left is $x$. So $5$ is the co-effi­cient of $x$.

Now take $-6xy$. Cover the $-6$. What is left is $xy$. So $-6$ is the co-effi­cient of $xy$. The minus sign goes with the $6$.

This rule works both ways round. Cover the $x$ in $5x$ instead. What is left is $5$. So $x$ is the co-effi­cient of $5$. That sounds odd. Here is why it hap­pens. A term is only parts mul­ti­plied together. Cover any part and the rest is still a prod­uct. So the rule can­not tell a num­ber from a let­ter. It knows parts, and noth­ing more.

In every­day work peo­ple mean less than that. The num­ber is the part that tells you how many. That is the part peo­ple usu­ally want. So when some­one asks for the co-effi­cient, they mean the num­ber in front. In $5x$ it is $5$. In $-6xy$ it is $-6$.

Remem­ber. Cover one fac­tor and the rest is a prod­uct. The fac­tor you cov­ered is the co-effi­cient of that prod­uct. In every­day use peo­ple mean the num­ber in front. Take its sign with it.

Worked exam­ple: hid­den co-effi­cients

Some terms seem to have no num­ber in front. Look at $x$. It means one lot of $x$, so $x = 1 \times x$, and the co-effi­cient of $x$ is $1$. We sim­ply do not write the $1$.

In the same way, $-y = (-1) \times y$. So the co-effi­cient of $y$ in $-y$ is $-1$.

And in $\displaystyle -\dfrac{ab}{3}$, the num­ber in front is $\displaystyle -\dfrac{1}{3}$, because $\displaystyle -\dfrac{ab}{3} = -\dfrac{1}{3} \times ab$. So the co-effi­cient of $ab$ is $\displaystyle -\dfrac{1}{3}$.

Find the co-effi­cient

Write the num­ber in front of each one. That is the co-effi­cient of the rest.

Now the other way round

Cover the num­ber in front. Write down what is left. That is the co-effi­cient of the num­ber you cov­ered.

All three words together

1. A term is one amount, held together by mul­ti­ply­ing or divid­ing.
2. A fac­tor is a part that is mul­ti­plied to make a prod­uct.
3. A co-effi­cient is one fac­tor, named for the prod­uct of the other fac­tors. In every­day use it means the num­ber in front.

Name these three parts when you meet a piece of alge­bra. Later rules are writ­ten with these same three words. Once you can name the parts, you can read the rule.

Com­mon mis­takes

• Split­ting a term at a mul­ti­pli­ca­tion. $5xy$ is one term, not three.
• Leav­ing the minus sign behind. In $3 - 2x$, the sec­ond term is $-2x$, and its co-effi­cient is $-2$.
• Say­ing $x$ has no co-effi­cient. Its co-effi­cient is $1$, and the co-effi­cient in $-x$ is $-1$.
• Think­ing a let­ter is short for a word, such as $w$ for "weight". The let­ter only holds the place of a num­ber.
• For­get­ting that a num­ber on its own, such as $6$, is a term too.

Key terms

Vari­able

A let­ter that stands for a num­ber, which may change.

Con­stant

A num­ber that never changes, such as $5$ or $-3$.

Term

One sin­gle amount, made of con­stants and vari­ables joined by mul­ti­ply­ing or divid­ing.

Expres­sion

One or more terms joined by plus and minus, such as $3 + 2x$.

Prod­uct

The result of mul­ti­ply­ing things together.

Fac­tor

A part that is mul­ti­plied to make a prod­uct.

Co-effi­cient

One fac­tor, named for the prod­uct of the oth­ers; in every­day use, the num­ber in front.

Answers

Count the terms

1. $6x$: $1$ term.
2. $3 + 2x$: $2$ terms, $3$ and $2x$.
3. $5xy$: $1$ term.
4. $a + b - 4$: $3$ terms, $a$, $b$ and $-4$.

Find the fac­tors

1. $7xy$: $7, \; x, \; y$
2. $3ab$: $3, \; a, \; b$
3. $-4m$: $-4, \; m$
4. $12pq$: $12, \; p, \; q$

Find the co-effi­cient

1. $9x$: $9$
2. $-2y$: $-2$
3. $15mn$: $15$
4. $-7ab$: $-7$

Now the other way round

1. $15mn$: $mn$ is the co-effi­cient of $15$.
2. $9x$: $x$ is the co-effi­cient of $9$.
3. $-7ab$: $ab$ is the co-effi­cient of $-7$.