Terms, monomials and polynomials

Every chap­ter that fol­lows in alge­bra, from fac­tori­sa­tion to qua­dratic equa­tions, is writ­ten in the lan­guage of poly­no­mi­als. Before we divide them, fac­torise them or find their zeros, we need the words: what a term is, which expres­sions count as poly­no­mi­als and which do not, and how to read off the degree. This les­son builds that vocab­u­lary care­fully, because a sin­gle wrong idea here (for exam­ple, think­ing $\displaystyle \frac{6}{x}$ is a poly­no­mial) causes mis­takes in every later les­son.

Terms#

A term is a con­stant on its own, a vari­able on its own, or con­stants and vari­ables com­bined by mul­ti­pli­ca­tion or divi­sion.

So each of these is a sin­gle term:

$$\displaystyle 6, \quad x, \quad 6x, \quad \frac{6}{x}, \quad 4x^2y, \quad \frac{4x^3y}{3}$$

Terms are sep­a­rated from one another by $+$ and $-$ signs. The expres­sion $3x^2 - 5x + 7$ has three terms: $3x^2$, $-5x$ and $7$. The sign in front belongs to the term, so the sec­ond term is $-5x$, not $5x$.

In a term such as $-5x$, the num­ber $-5$ is the coef­fi­cient and $x$ is the vari­able part. A term with no vari­able at all, like $7$, is called a con­stant term.

Mono­mi­als#

A mono­mial is an expres­sion with exactly one term in which every power of every vari­able is a non-neg­a­tive inte­ger, that is, one of $0, 1, 2, 3, 4, \dots$

$$\displaystyle 6x^2y, \quad 6x^2y^3, \quad 4x^2y^2, \quad -\frac{2}{3}xy, \quad 7y$$

are all mono­mi­als. Now look at two terms that are not mono­mi­als:

$$\displaystyle \frac{6x^2}{y} = 6x^2y^{-1}, \qquad 6\sqrt{x} = 6x^{\frac{1}{2}}$$

The first has the power $-1$ on $y$, which is neg­a­tive. The sec­ond has the power $\displaystyle \frac{1}{2}$ on $x$, which is not an inte­ger. Both are terms, but nei­ther is a mono­mial.

What about a plain num­ber such as $6$? Since $x^0 = 1$, we can write

$$6 = 6 \times 1 = 6x^0$$

The power of $x$ is $0$, which is allowed, so a con­stant is a mono­mial too.

Remem­ber. A term only has to be built by mul­ti­ply­ing and divid­ing. A mono­mial is stricter: no vari­able may sit in a denom­i­na­tor and no vari­able may sit under a root sign.

Bino­mi­als, tri­no­mi­als and poly­no­mi­als#

Mono­mi­als are joined by addi­tion and sub­trac­tion to make big­ger expres­sions, and we name them by how many mono­mi­als they con­tain.

• A bino­mial con­tains two mono­mi­als: $2x^2y + 5xy$, $\;6 + x$.
• A tri­no­mial con­tains three mono­mi­als: $6 + x - x^2$, $\;4x^3y + 5xy + 7$.
• A poly­no­mial con­tains one or more mono­mi­als: $6$, $\;6 + x$, $\;6 + x + x^2$, $\;x^4 + x^3y + 7x + 8$.

So every mono­mial, bino­mial and tri­no­mial is a poly­no­mial. The word "poly­no­mial" sim­ply means "many terms", and it includes the case of a sin­gle term.

A poly­no­mial may use one vari­able, as in $t^3 - 3t^2 + 4t - 5$, or sev­eral, as in $x^2 - xy + 7y^2$ (two vari­ables) or $xy + yz + zx$ (three vari­ables). Most of this chap­ter works with poly­no­mi­als in one vari­able, usu­ally $x$.

Exam­ple 1#

Decide which of these are poly­no­mi­als: (a) $3x^2 - 4x + 15$, (b) $\displaystyle x - \frac{4}{x}$, (c) $3\sqrt{x} + 2x$, (d) $x^2 + y^3 + t^{50}$.

(a) Every power is a whole num­ber, so it is a poly­no­mial in one vari­able.

(b) $\displaystyle \frac{4}{x} = 4x^{-1}$ has a neg­a­tive power, so this is not a poly­no­mial.

(c) $\displaystyle 3\sqrt{x} = 3x^{\frac{1}{2}}$ has a frac­tional power, so this is not a poly­no­mial.

(d) The pow­ers $2$, $3$ and $50$ are whole num­bers, so it is a poly­no­mial, in three vari­ables.

Degree of a mono­mial#

The degree of a mono­mial is the sum of the expo­nents of its vari­ables.

Exam­ple 2#

Find the degree of $4x^2yz^3$.

The expo­nents are $2$ on $x$, $1$ on $y$ (since $y = y^1$) and $3$ on $z$.

$$2 + 1 + 3 = 6$$

The degree is $6$. Notice that the coef­fi­cient $4$ plays no part in the degree.

A non-zero con­stant such as $7$ can be writ­ten $7x^0$, so its degree is $0$.

Degree of a poly­no­mial#

The degree of a poly­no­mial is the great­est degree among all its terms. Find the degree of each term sep­a­rately, then pick the largest.

Exam­ple 3#

Find the degree of $10x^2y + x^3y^2 - xy^6 + 2x + 7$.

Work term by term:

The great­est of $3, 5, 7, 1, 0$ is $7$, so the degree of the poly­no­mial is $7$.

A com­mon slip is to choose the term with the biggest sin­gle power. Here $x^3y^2$ has the biggest power of $x$, but the term $-xy^6$ has the larger total.

Exam­ple 4#

Find the degrees of (a) $x - 1$ and (b) $10x^9 - 5x^2 + 1$.

(a) The terms have degrees $1$ and $0$. The great­est is $1$, so the degree is $1$.

(b) The terms have degrees $9$, $2$ and $0$. The degree is $9$.

Exam­ple 5#

Write the coef­fi­cient of $x^2$ in (a) $17 - 2x + 7x^2$ and (b) $9 - 12x + x^3$.

(a) The term in $x^2$ is $7x^2$, so the coef­fi­cient is $7$.

(b) There is no $x^2$ term at all. We can think of it as $0x^2$, so the coef­fi­cient of $x^2$ is $0$.

Prac­tice#

1. Which of these are mono­mi­als? $\displaystyle 7x^3, \; \frac{5}{x}, \; -2ab^2, \; 4\sqrt{y}, \; 9$
2. Name each as a mono­mial, bino­mial or tri­no­mial: (a) $3x - 7$ (b) $5x^2yz$ (c) $x^2 - 7x + 12$ (d) $xy + yz + zx$
3. Find the degree of each mono­mial: (a) $9x^4$ (b) $-3a^2b^3$ (c) $5xyz$ (d) $11$
4. Find the degree of $7x^3 + 4x^2 - 3x + 12$.
5. Find the degree of $12 - x + 2x^3$.
6. Find the degree of $7x^3y^2 - 3x^2y + 5xy - 8$.
7. Find the degree of $4 + 2a^2b^4 - ab^3$.
8. Write the coef­fi­cient of $x^2$ in $\displaystyle \frac{\pi}{6}x^2 - 3x + 4$.
9. How many vari­ables does each poly­no­mial have? (a) $x^2 - xy + 7y^2$ (b) $t^3 - 3t^2 + 4t - 5$ (c) $xy + yz + zx$
10. The mono­mial $3x^ay^2$ has degree $7$. Find $a$.

Answers#