The degree of an expression

Once you can sort an expres­sion into mono­mi­als, bino­mi­als and poly­no­mi­als, the next thing to know about it is its degree. The degree tells you how fast 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 $2$ and 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#

Degree of a mono­mial. The degree of a mono­mial is the sum of the pow­ers of the vari­ables in it.

Look at $x^2y$. It means $x \times x \times y$: three let­ters mul­ti­plied together. Adding the pow­ers, $2 + 1 = 3$, does that count­ing for you. So the degree of $x^2y$ is $3$.

A let­ter with no power writ­ten has power $1$, so $x$ means $x^1$.

The num­ber in front never changes the degree. In $7x^3yz^2$, the $7$ is the co-effi­cient. It is not a vari­able, so it adds noth­ing.

Exam­ple 1#

Find the degree of $-4a^2b^3c^4$.

Ignore the $-4$, which is the co-effi­cient. The pow­ers are $2$, $3$ and $4$, and $2 + 3 + 4 = 9$. The degree is $9$. The minus sign changes noth­ing, because it belongs to the num­ber.

Why the degree of 5 is 0#

A plain num­ber holds no let­ters at all, yet it can still be writ­ten with one. Count the pow­ers of $x$ down­wards, divid­ing by $x$ at each step: $x^3 \div x = x^2$ and $x^2 \div x = x^1$. One more step gives $x^0$, and that step is $x \div x$, which is $1$. So $x^0 = 1$, for any $x$ other than $0$.

Now $5 = 5 \times 1 = 5x^0$. The only power in sight is $0$, so the degree of $5$ is $0$. Every non-zero num­ber works the same way.

Why the degree of 0 is unde­fined#

Zero behaves dif­fer­ently. We can write

$$0 = 0 \times x^0 = 0 \times x^1 = 0 \times x^2 = \cdots$$

Zero can be writ­ten with any power we like, and it stays $0$ each time. A degree has to be one fixed num­ber, and here every num­ber works equally well, so there is no way to choose. That is why the degree of $0$ is unde­fined.

The degree of a poly­no­mial#

Degree of a poly­no­mial. The great­est degree among the terms of a poly­no­mial is called the degree of the poly­no­mial.

So find the degree of each term, then pick the largest.

Exam­ple 2#

Find the degree of $4x^2y + 5x^2y^5 - 5x - 3$.

The great­est of $3$, $7$, $1$ and $0$ is $7$. So the degree of the poly­no­mial is $7$.

Exam­ple 3#

Find the degree of $x^5 + 4x^3 + 3x - 5$, and of $2x + 5$.

In the first, the term degrees are $5$, $3$, $1$ and $0$, so the degree is $5$. In the sec­ond, they are $1$ and $0$, so the degree is $1$.

Why the great­est, and not the total?#

Take the poly­no­mial $x^5y^3 + 4x^2y^4 + 6xy + 9$. Its term degrees are $8$, $6$, $2$ and $0$, so its degree is $8$. To see why the biggest degree mat­ters most, make both let­ters large at the same time. Put $x = 10$ and $y = 10$.

The degree $8$ term is $25$ times the next one, and it tow­ers over the other two. Make the let­ters larger still and the gap only grows. The term with the great­est degree decides how the whole poly­no­mial behaves, which is why it gives the degree.

The terms may be writ­ten in any order. You are look­ing for the biggest degree, not the first one writ­ten. For exam­ple, $3 - 2x + 5x^2$ has degree $2$, even though $5x^2$ comes last.

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

Exam­ple 4#

Find the degree of $5xy + yz + zx$.

Each term has two vari­ables with power $1$, so each has degree $2$. The degree is $2$.

Exam­ple 5#

Find the degree of $x^2y + yx + yx^3$.

The term degrees are $2 + 1 = 3$, $1 + 1 = 2$ and $1 + 3 = 4$. The great­est is $4$, so the degree is $4$.

Com­mon mis­takes#

• Count­ing the co-effi­cient. The degree of $9m^2n^3$ 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. $x^2y^3$ has degree $5$, not $6$.
• Adding the degrees of all the terms of a poly­no­mial. Take the great­est one instead.
• Say­ing the degree of $0$ is $0$. It is unde­fined; only non-zero num­bers have degree $0$.

Prac­tice#

1. Find the degree of $a^4b$.
2. Find the degree of $pqr$.
3. Find the degree of $9m^2n^3$.
4. Find the degree of $12$.
5. Find the degree of $1 - 3x$.
6. Find the degree of $x^2y^3 + x^4 + 5$.
7. Find the degree of $a^3b^2c + ab + 7$.
8. Find the degree of $4p^3q - p^2q^2r + 11$.
9. Find the degree of $x^4 + 3x - x^4 + 2$.

Answers#