What an algebraic expression is

What this lesson is about

When a shopkeeper says a pen costs ₹5 more than a pencil, and you do not yet know the price of the pencil, you can still write the price of the pen: $x + 5$ . That short line is an algebraic expression. Expressions like it appear in formulas for area and perimeter, in word problems, and in every equation you will solve.

This lesson gives a name to the thing you write when you put terms together. It then sorts those things into five types, so that when a question says binomial or polynomial you know exactly what it means.

This lesson gives a name to the thing you write when you put terms together. It then sorts those things into five types.

You already know what a term is. The Early Years course "Introduction to Algebra" defines it in full.

The fundamental operations

Four operations are called the fundamental operations. They are adding, taking away, multiplying and dividing.

You already use all four on plain numbers.

$3 + 4 = 7$	$3 \times 4 = 12$
$9 - 4 = 5$	$12 \div 4 = 3$

The same four signs are used with letters. Nothing new is added here. Only the numbers are replaced by letters.

An algebraic expression is one or more terms joined by $+$ or $-$ signs. Multiplying and dividing do not join terms. They stay inside one term, which is what the word term means.

The chapter says it this way: the combination of terms obtained by the fundamental operations.

Here are three algebraic expressions.

5x + 3

7ab - 2b

\displaystyle \frac{x}{4} + 9

Look at $5x + 3$ . Its terms are $5x$ and $3$ . A plus sign joins them, so the whole line is an expression.

One term on its own says very little. Joined terms carry a whole rule: $5x + 3$ says take a number, multiply it by $5$ , then add $3$ .

Reading an expression term by term

Take $7ab - 2b$ . Split it wherever a plus or minus sign joins two parts. The terms are $7ab$ and $-2b$ ; the minus sign travels with the term after it. Inside $7ab$ the operation is multiplying, so $7$ , $a$ and $b$ stay together as one term.

Now take $\displaystyle \frac{x}{4} + 9$ . The division sits inside the first term, so $\displaystyle \frac{x}{4}$ is one term, and $9$ is the second. Two terms, joined by a plus sign.

Writing an expression from words

Words turn into expressions one operation at a time. "Three times a number, less two" becomes $3n - 2$ . "The sum of a number and its square" becomes $y + y^{2}$ . "Half of a number, added to seven" becomes $\displaystyle \frac{p}{2} + 7$ . In each one, check that you can point to the terms and to the signs that join them.

Remember. An algebraic expression is terms joined by the fundamental operations.

The five types

Expressions are sorted by how many parts they hold. Four of the five names are built from monomials, so they carry the power rule with them. Only the multinomial is built from terms.

Monomial

An expression with only one term is called a monomial. $7x^{2}y$ is one. Every power in it must be a whole number: $0$ , $1$ , $2$ , $3$ and so on.

In $7x^{2}y$ the power of $x$ is $2$ and the power of $y$ is $1$ . Both are whole numbers, so it passes.

A variable is a letter standing in for a number. The formal name for whole numbers like these is the non-negative integers.

A power below zero is shut out. A power that is a fraction is shut out too.

The number in front of the letter may still be a fraction. $\displaystyle \frac{x}{4}$ is a monomial, because the power of $x$ is $1$ .

Here is why the powers are ruled this way. A whole number power just counts how many times the letter is multiplied by itself.

What a whole number power means

$\begin{aligned}&x^{3} \\ &= x \times x \times x\end{aligned}$

Now look at $x^{-1}$ . Its power is $-1$ , which sits below zero. It is not a monomial.

You cannot write $x^{-1}$ as a row of $x$ s multiplied together. It means $\displaystyle \frac{1}{x}$ , which is a division.

The test is about the LETTERS, not the numbers. Every letter's power must be a whole number. A number underneath, as in $\displaystyle \frac{x}{4}$ , does not matter, because that is just $\displaystyle \frac{1}{4}$ of $x$ and the power of $x$ is still $1$ . But a LETTER underneath, as in $\displaystyle \frac{3}{x}$ , means the power of $x$ is $-1$ , and that is not a whole number.

Look at $\sqrt{x}$ as well. A square root is the power $\displaystyle \frac{1}{2}$ . A half is not a whole number, so $\sqrt{x}$ is not a monomial.

Half an $x$ multiplied by itself is not a row of $x$ s. So $\sqrt{x}$ falls outside as well.

Remember. One term is not enough. A monomial also needs every power to be a whole number.

Binomial

An expression containing two monomials is called a binomial. $3x + 5$ is a binomial. So is $7ab - 2c$ .

Trinomial

An expression containing three monomials is called a trinomial. $x^{2} + 4x + 4$ is a trinomial. So is $2a + 3b - c$ .

Polynomial

An expression containing one or more monomials is called a polynomial. $x^{3} + 2x^{2} + x + 6$ is a polynomial. It holds four monomials.

Polynomial is the wide name. Every monomial is a polynomial. Every binomial and every trinomial is one as well.

Multinomial

An expression containing one or more terms is called a multinomial. $2p - 9$ is a multinomial.

Read the last two definitions side by side. A polynomial is built from monomials. A multinomial is built from terms.

That one word is the whole difference. A term may divide by a letter, which puts the power below zero. A monomial may not.

Remember. All polynomials are multinomials. Not every multinomial is a polynomial.

Start with the first half. A monomial is a term, so an expression made of monomials is made of terms.

The second half needs one example. Test $\displaystyle \frac{5}{x} + 2$ against both definitions.

Count the terms. $\displaystyle \frac{5}{x}$ is one term and $2$ is another.
Two terms is one or more terms, so this is a multinomial.
Now test the terms. Write $\displaystyle \frac{5}{x}$ as $5x^{-1}$ .
The power is $-1$ , which is below zero. That term is not a monomial.
A polynomial is made of monomials only. So $\displaystyle \frac{5}{x} + 2$ is not a polynomial.

One bad term is enough to spoil it. Count the terms first, then check each power.

Nested boxes: multinomials contain polynomials, which contain monomial 7x squared y, binomial 3x + 5 and trinomial x squared + 4x + 4; 5/x + 2 and root y + 3 sit outside. — Every polynomial sits inside the larger family of multinomials. Expressions with a letter underneath or under a root stay outside the polynomial box.

A worked sorting example

Sort $4x^{2} - 3xy + y^{2} - 7$ .

Split at the signs. The terms are $4x^{2}$ , $-3xy$ , $y^{2}$ and $-7$ . That is four terms.
Check each power. In $4x^{2}$ the power is $2$ . In $-3xy$ both powers are $1$ . In $y^{2}$ the power is $2$ . The number $-7$ has no letter at all. Every power is a whole number, so all four terms are monomials.
Four monomials is more than three, so it is not a trinomial. It is a polynomial, and therefore a multinomial as well.

Type	What it holds	Example
Monomial	One term, every power a whole number	$7x^{2}y$
Binomial	Two monomials	$3x + 5$
Trinomial	Three monomials	$x^{2} + 4x + 4$
Polynomial	One or more monomials	$x^{3} + 2x^{2} + x + 6$
Multinomial	One or more terms	$\displaystyle \frac{5}{x} + 2$

The first four rows count monomials, so every power must be a whole number. The last row counts terms, so it does not.

Name the type: $9y$
Name the type: $4a + 7$
Name the type: $x^{2} + 5x + 6$
Is $8m^{3}n$ a monomial? Say why.
Is $\sqrt{y} + 3$ a polynomial? Say why.
Write a trinomial of your own.

Question	Answer	Question	Answer
$9y$	A monomial. It is one term and the power of $y$ is $1$ .	$8m^{3}n$	Yes. It is one term, and the powers $3$ and $1$ are whole numbers.
$4a + 7$	A binomial. It holds the two monomials $4a$ and $7$ .	$\sqrt{y} + 3$	No. $\sqrt{y}$ has the power $\displaystyle \frac{1}{2}$ , which is not a whole number. So $\sqrt{y}$ is not a monomial. A polynomial is made of monomials only, so this is not a polynomial.
$x^{2} + 5x + 6$	A trinomial. It holds three monomials.	Your own trinomial	Any three monomials joined by $+$ or $-$ , such as $2p + 3q - 5$ .

Common mistakes

Counting factors as terms. $7ab$ is one term, not three, because multiplying does not separate terms.
Calling $\displaystyle \frac{3}{x}$ a monomial. A letter underneath means the power is $-1$ .
Thinking a fraction in front of a letter spoils a monomial. $\displaystyle \frac{x}{4}$ is a monomial, because the power of $x$ is $1$ .
Forgetting that a monomial is also a polynomial, and that every polynomial is also a multinomial.
Leaving the minus sign behind when splitting an expression into terms.

Key terms

Fundamental operations: Adding, taking away, multiplying and dividing.
Variable: A letter standing in for a number.
Algebraic expression: One or more terms joined by plus or minus signs.
Monomial: An expression with one term in which every power of a letter is a whole number.
Binomial: An expression containing two monomials.
Trinomial: An expression containing three monomials.
Polynomial: An expression containing one or more monomials.
Multinomial: An expression containing one or more terms, whatever their powers.

Answers

$9y$ is a monomial: one term, and the power of $y$ is $1$ .
$4a + 7$ is a binomial: it holds the two monomials $4a$ and $7$ .
$x^{2} + 5x + 6$ is a trinomial: it holds three monomials.
Yes. $8m^{3}n$ is one term, and the powers $3$ and $1$ are whole numbers.
No. $\sqrt{y}$ has the power $\displaystyle \frac{1}{2}$ , which is not a whole number, so it is not a monomial. The expression is a multinomial but not a polynomial.
Model answer: $2p + 3q - 5$ . Any three monomials joined by $+$ or $-$ will do.

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