The degree of an expression

Terms, fac­tors and co-effi­cients are taught in the Early Years course Intro­duc­tion to Alge­bra. This les­son uses them and does not teach them again.

An alge­braic expres­sion is terms joined by plus or minus signs, as What an alge­braic expres­sion is sets out. One exam­ple is $3x + 5$.

The let­ters in a term are called vari­ables. This les­son sim­ply calls them let­ters.

Here you meet one new num­ber: the degree.

The degree tells you how "big" an expres­sion grows when its let­ters grow. It is the first thing you look at when you name an expres­sion: lin­ear expres­sions have degree $1$, qua­dratic ones degree $2$, cubic ones degree $3$. You will use it when you solve equa­tions, draw graphs and divide poly­no­mi­als in later classes.

The degree of a mono­mial

A power counts how many times a let­ter is mul­ti­plied. In $x^{2}$ the power is $2$.

A mono­mial is an expres­sion with only one term. Every power in it must be a whole num­ber.

Whole num­bers here are $0$, $1$, $2$ and so on.

So $\displaystyle \frac{3}{x}$ is not a mono­mial. The let­ter sits under a divi­sion line, and that is not a whole-num­ber power.

$\sqrt{x}$ is not a mono­mial either. A root sign means the power $\displaystyle \frac{1}{2}$, which is not a whole num­ber.

$7x^{2}y$ is a mono­mial. It is one term, and both pow­ers are whole num­bers.

That rule mat­ters. It is why every degree in this les­son is a whole num­ber.

Look at $x^{2}y$. The let­ter $x$ appears twice. The let­ter $y$ appears once.

So the term means $x \times x \times y$. That is three let­ters mul­ti­plied together.

Count­ing those let­ters gives $3$. The degree of $x^{2}y$ is $3$.

Degree of $x^{2}y$

$$\begin{aligned}&2 + 1 \\ &= 3\end{aligned}$$

Adding the pow­ers does that count­ing for you. That is the whole rule.

Now take $xy$. Nei­ther let­ter shows a power.

A let­ter on its own means the power $1$. Here $x$ means $x^{1}$, one $x$ mul­ti­plied once.

So both pow­ers are $1$.

Degree of $xy$

$$\begin{aligned}&1 + 1 \\ &= 2\end{aligned}$$

Take $x^{3}yz^{2}$. The pow­ers are $3$, $1$ and $2$.

Degree of $x^{3}yz^{2}$

$$\begin{aligned}&3 + 1 + 2 \\ &= 6\end{aligned}$$

Take $x^{2}yz$. The pow­ers are $2$, $1$ and $1$.

Degree of $x^{2}yz$

$$\begin{aligned}&2 + 1 + 1 \\ &= 4\end{aligned}$$

The num­ber in front never changes the degree.

In $7x^{2}y$ the $7$ is the co-effi­cient of $x^{2}y$. It is not a let­ter, so it adds noth­ing.

The degree of $7x^{2}y$ is still $3$.

One more worked exam­ple. Find the degree of $-4a^{2}b^{3}c^{4}$. Step 1: ignore the $-4$, which is the co-effi­cient. Step 2: the pow­ers are $2$, $3$ and $4$. Step 3: add them.

$$\begin{aligned}&2 + 3 + 4 \\ &= 9\end{aligned}$$

So the degree is $9$. A minus sign in front changes noth­ing either; it belongs to the num­ber.

Remem­ber. To find the degree of a mono­mial, add the pow­ers of its let­ters. Leave the num­ber in front out.

Why the degree of $5$ is $0$

A plain num­ber holds no let­ters at all.

You can still write one with a let­ter in it. First look at $x^{0}$.

Count the pow­ers of $x$ down­wards. Each step down divides by $x$.

$x^{3} \div x = x^{2}$, and $x^{2} \div x = x^{1}$.

One more step gives $x^{0}$. That step is $x \div x$, which is $1$.

So $x^{0} = 1$. This needs $x$ to be some num­ber other than $0$.

Now $5 = 5 \times x^{0}$, because $5 \times 1 = 5$.

The only power in sight is $0$. So the degree of $5$ is $0$.

Every plain num­ber works the same way.

Why the degree of $0$ is unde­fined

Zero behaves dif­fer­ently from other num­bers.

Look at $0 \times x^{2} = 0$. Look at $0 \times x^{7} = 0$ as well.

You may write $0$ with any power you like. The answer stays $0$ each time.

A degree has to be one fixed num­ber for a term.

Here every num­ber works equally well, so there is no way to choose.

The degree of $0$ is unde­fined.

Remem­ber. The degree of $5$ is $0$. The degree of $0$ is unde­fined.

The degree of a poly­no­mial

A poly­no­mial is an expres­sion con­tain­ing one or more mono­mi­als.

So a sin­gle mono­mial such as $7x^{2}y$ is a poly­no­mial as well.

Two mono­mi­als make a bino­mial. Three make a tri­no­mial. Those names come with the types of expres­sions.

Take the poly­no­mial $x^{5}y^{3} + 4x^{2}y^{4} + 6xy + 9$.

Its terms are joined by $+$ signs. Each term has its own degree.

Work out the degree of each term first.

Now pick the great­est of those four degrees.

Great­est of the four term degrees

$$\begin{aligned}&\text{greatest of } 8,\ 6,\ 2,\ 0 \\ &= 8\end{aligned}$$

So the degree of that poly­no­mial is $8$.

The great­est degree among the terms is the degree of the poly­no­mial.

Why the great­est? Make every let­ter large at the same time.

Put $x = 10$ and $y = 10$ into the four terms.

The degree $8$ term is $25$ times the next one. It dwarfs the other three.

Make both let­ters larger still and that gap grows.

Now change only one let­ter and this can fail.

Hold $x$ at $1$ and let $y$ grow to $10$.

The first term is then $1\,000$ and the sec­ond is $40\,000$.

The degree $6$ term is the larger one there.

So the degree adds up all the pow­ers of a term. It is the right mea­sure when the let­ters grow together.

The terms may be writ­ten in any order. The degree does not change.

You are hunt­ing for the biggest one, not the first one.

Remem­ber. Find the degree of every term. The largest of them is the degree of the poly­no­mial.

Two more poly­no­mi­als

Exam­ple A. Find the degree of $3 - 2x + 5x^{2}$. The terms have degrees $0$, $1$ and $2$. The great­est is $2$, so this is a poly­no­mial of degree $2$. Notice the biggest term was writ­ten last.

Exam­ple B. Find the degree of $4p^{3}q - p^{2}q^{2}r + 11$.

The great­est of $4$, $5$ and $0$ is $5$. So the degree is $5$. The minus sign in front of the sec­ond term does not mat­ter.

One cau­tion: join any like terms first. In $x^{3} + 2x - x^{3}$ the two $x^{3}$ terms can­cel, leav­ing $2x$. Its degree is $1$, not $3$.

Prac­tice

1. Find the degree of $a^{4}b$.
2. Find the degree of $pqr$.
3. Find the degree of $9m^{2}n^{3}$.
4. Find the degree of $12$.
5. Is $\displaystyle \frac{5}{x}$ a mono­mial? Give your rea­son.
6. Find the degree of $x^{2}y^{3} + x^{4} + 5$.
7. Find the degree of $a^{3}b^{2}c + ab + 7$.

Com­mon mis­takes

• Count­ing the co-effi­cient. The degree of $9m^{2}n^{3}$ is $5$, not $14$.
• For­get­ting that a bare let­ter has power $1$. The degree of $pqr$ is $3$, not $0$.
• Mul­ti­ply­ing the pow­ers instead of adding them. $x^{2}y^{3}$ has degree $5$, not $6$.
• Adding the degrees of all the terms of a poly­no­mial. Take the great­est one instead.
• Tak­ing the degree of the first term writ­ten. The largest term may be any­where.
• Say­ing the degree of $0$ is $0$. It is unde­fined; only non-zero num­bers have degree $0$.

Key terms

Power

The small raised num­ber that counts how many times a let­ter is mul­ti­plied.

Mono­mial

An expres­sion with one term, in which every power is a whole num­ber.

Degree of a mono­mial

The sum of the pow­ers of its let­ters.

Poly­no­mial

An expres­sion con­tain­ing one or more mono­mi­als.

Degree of a poly­no­mial

The great­est degree among its terms.

Co-effi­cient

The num­ber in front of the let­ters; it never changes the degree.

Con­stant

A plain num­ber; a non-zero con­stant has degree $0$.

Answers

1. $a^{4}b$: $4 + 1 = 5$.
2. $pqr$: $1 + 1 + 1 = 3$.
3. $9m^{2}n^{3}$: $2 + 3 = 5$; the $9$ is ignored.
4. $12$: degree $0$, because it is a non-zero plain num­ber.
5. No. $\displaystyle \dfrac{5}{x}$ is $5x^{-1}$; the let­ter is under a divi­sion line, so its power is not a whole num­ber.
6. Term degrees $5$, $4$ and $0$; the degree is $5$.
7. Term degrees $6$, $2$ and $0$; the degree is $6$.