FOUNDATIONS 02

Algebraic Expressions

Algebra uses symbols to describe quantities and relationships generally, rather than only one numerical case.

The main idea

Arithmetic calculates with known numbers. Algebra introduces symbols so the same rule can work for many possible values.

For example, if a taxi costs £3 plus £2 for each mile, the cost of travelling \(x\) miles is

\[\boxed{C=3+2x}.\]

Changing \(x\) changes the cost, but the relationship stays the same. This idea leads directly to formulas, functions, graphs and calculus.

1. Variables

A variable is a symbol representing a quantity whose value may change or may not yet be known.

\[x,\qquad y,\qquad t\]

In \(x+4\), the value of the expression changes when \(x\) changes.

\(x=1\Rightarrow x+4=5\)
\(x=3\Rightarrow x+4=7\)
\(x=10\Rightarrow x+4=14\)

One expression represents many numerical cases.

2. Constants and coefficients

A constant is a fixed value. A coefficient is a number multiplying a variable.

In \(3x+5\), \(3\) is the coefficient of \(x\), while \(5\) is a constant.

\(\underbrace{3}_{\text{coefficient}}\underbrace{x}_{\text{variable}}+\underbrace{5}_{\text{constant}}\)

The coefficient controls how much the term changes when the variable changes.

3. Terms

Terms are the parts of an expression separated by addition or subtraction.

\[4x^2-3x+6\]

This expression has three terms: \(4x^2\), \(-3x\), and \(6\).

\(\boxed{4x^2}\quad-\quad\boxed{3x}\quad+\quad\boxed{6}\)

quadratic termlinear termconstant term

4. Like terms

Like terms have the same variable part, including the same powers. Their coefficients can therefore be combined.

\[3x+5x=8x.\]

But \(3x\) and \(5x^2\) are not like terms, because \(x\) and \(x^2\) are different variable parts.

\(3x+5x=(3+5)x=8x\)
\(3x+5x^2\) cannot be combined

5. Evaluating an expression

To evaluate an expression, substitute a particular value for its variable and calculate the result.

For \(2x+1\), if \(x=3\),

\[2(3)+1=7.\]

general rule
\(2x+1\)
↓ \(x=3\)
particular value
\(7\)

6. Expression versus equation

An expression represents a quantity. An equation states that two expressions have equal values.

The equals sign changes the mathematical object and what we can do with it.

Expression\[3x+2\]represents a quantity

Equation\[3x+2=11\]states an equality

7. Building an expression from words

Algebra becomes useful when a verbal relationship is translated into symbols.

If a rectangle has width \(x\) and length \(x+3\), its area is

\[A=x(x+3).\]

Key takeaways

A variable can change; a constant stays fixed; a coefficient multiplies a variable; and terms form an expression. Substitution turns a general expression into a numerical value, while an equation states that two expressions are equal.

Why this matters for calculus

Calculus studies how quantities change. Before differentiating or integrating a formula, we need to recognise its variables, constants, coefficients, terms and algebraic structure. For example, understanding \(3x^2+2x-1\) algebraically is the first step toward understanding how it changes with \(x\).