Algebraic Expressions

What you will learn

• Use the words term, coefficient, constant and like terms.
• Write expressions with algebra shorthand.
• See where substitution, expanding and factorising fit.

Before you start: you should know what a letter standing for a number means. See Number Sequences and Rules.

The idea

An algebraic expression combines numbers, letters and operations, such as $3x + 7$ or $a^2 - 2ab$. The letters are variables: they stand for numbers that can change. An expression has no equals sign. It is not a statement to solve but a recipe that produces a value once you know the variables.

The vocabulary

Take the expression $4x^2 - 3x + 9$.

• It has three terms, separated by $+$ and $-$ signs: $4x^2$, $-3x$ and $9$.
• The number in front of a variable is its coefficient: $4$ is the coefficient of $x^2$ and $-3$ is the coefficient of $x$.
• A term with no variable, here $9$, is a constant.

Writing conventions

Algebra has some shorthand. $3 \times x$ is written $3x$, $x \times x$ is written $x^2$, and $a \div b$ is usually written $\frac{a}{b}$. We put the number first: $5y$, not $y5$.

Like terms

Like terms have exactly the same letters raised to the same powers. $5x$ and $-2x$ are like terms; $5x$ and $5x^2$ are not. Only like terms can be added or subtracted, just as you can add apples to apples but not apples to oranges.

The next lessons teach each skill in depth: substitution, like terms, expanding and factorising.

Worked examples

Example 1: Simplifying

Simplify $6a + 3b - 2a + 5 - b$.

1. Group the like terms: $(6a - 2a) + (3b - b) + 5$.
2. Combine each group: $4a + 2b + 5$.

Answer: $4a + 2b + 5$

Example 2: Substitution

Find the value of $2x^2 - 3y$ when $x = -4$ and $y = 5$.

1. Replace each letter with its value, using brackets: $2(-4)^2 - 3(5)$.
2. Powers first: $(-4)^2 = 16$, so this is $2 \times 16 - 15$.
3. Then multiply and subtract: $32 - 15 = 17$.

Answer: $17$

Example 3: Expanding and factorising

Expand $(x + 3)(x - 5)$, then factorise $6x^2 + 9x$.

1. To expand, multiply every term in the first bracket by every term in the second: $x \cdot x + x \cdot (-5) + 3 \cdot x + 3 \cdot (-5)$.
2. Simplify: $x^2 - 5x + 3x - 15 = x^2 - 2x - 15$.
3. To factorise $6x^2 + 9x$, find the highest common factor of both terms. That is $3x$.
4. Divide each term by $3x$ and write the result in a bracket: $3x(2x + 3)$.

Answer: $x^2 - 2x - 15$ and $3x(2x + 3)$

Factorising is expanding in reverse. You can always check a factorisation by expanding it again.

Why it matters

Expressions let you describe a whole family of calculations at once. A phone plan that charges £12 a month plus 8p per minute costs $12 + 0.08m$ pounds, whatever the number of minutes $m$. A spreadsheet formula like =B2*1.2 is an algebraic expression in disguise.

Simplifying and factorising are the core moves used later to solve equations, sketch graphs and, in calculus, differentiate and integrate.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.

Question 1 of 8

How many terms are in $4x^2 - 3x + 9$?

Your answer

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Question 2 of 8

What is the coefficient of $x$ in $4x^2 - 3x + 9$?

Your answer

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Question 3 of 8

What is the constant term in $4x^2 - 3x + 9$?

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Question 4 of 8

Which two are like terms?

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$5x$ and $-2x$$5x$ and $5x^2$$3a$ and $3b$$4$ and $4x$

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Question 5 of 8

Simplify $6a + 3b - 2a + 5 - b$. Give the coefficient of $a$.

Your answer

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Question 6 of 8

Expand $3(x + 2)$. Give the constant term.

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Question 7 of 8

Factorise $2x + 6$ as $2(x + \ldots)$. What is the missing number?

Your answer

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Question 8 of 8

Find $2x + 1$ when $x = 5$.

Your answer

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