Algebra - Expressions Notes | Junior Cycle Mathematics

An algebraic expression brings together letters and numbers without an equals sign. In Higher Level exams, you must simplify terms, expand brackets up to cubics, factorise four distinct structures, divide polynomials, add algebraic fractions, and rearrange formulas.

Simplifying and Multiplying Expressions

Five € $x$ tickets plus a €3 fee cost € $(5x + 3)$ . A €28 total gives $5x + 3 = 28$ : both sides are equal.

Combine like terms: $4x^2$ and $7x^2$ give $11x^2$ ; $4x^2$ and $7x$ stay separate. When multiplying single terms, multiply coefficients; add powers only for the same base: $3x^2 \times 4x^3 = 12x^5$ .

To multiply brackets, use split-and-repeat:

Two linear brackets: $(2x + 3)(x - 5) = 2x(x - 5) + 3(x - 5) = 2x^2 - 7x - 15$ .
Linear by quadratic: $(2x - 1)(x^2 + 3x - 4) = 2x(x^2 + 3x - 4) - 1(x^2 + 3x - 4) = 2x^3 + 5x^2 - 11x + 4$ .

Substitute means replace each letter with its value; put negative values in brackets: for example, evaluating $3x^2 - 5x$ when $x = -2$ gives $3(-2)^2 - 5(-2) = 12 + 10 = 22$ .

The Four Methods of Factorising

Factorising writes an expression as a product of factors. Always check for a common factor first before using another method:

Method	Strategy	Example
Highest common factor	Divide out greatest shared number and letter	$6x^2 - 18x = 6x(x - 3)$
Grouping in pairs	Group four terms into pairs sharing a bracket	$3x(a + b) - 2(a + b) = (a + b)(3x - 2)$
Difference of two squares	Form $(ax - by)(ax + by)$	$25x^2 - 49y^2 = (5x - 7y)(5x + 7y)$
Trinomial ( $a = 1$ )	Find factors of $c$ that add to $b$	$x^2 - 5x - 14 = (x - 7)(x + 2)$
Trinomial ( $a > 1$ )	Find factors of $ac$ adding to $b$ ; split middle term	$2x^2 - 7x - 15 = (2x + 3)(x - 5)$

Algebraic Fractions and Long Division

Division by zero is undefined, so denominators and cancelled factors must be non-zero. When adding or subtracting fractions of the form $\frac{a}{bx + c}$ , create a single fraction using a common denominator (here $x \neq -2, 1$ ):

\frac{3}{x + 2} - \frac{2}{x - 1} = \frac{3(x - 1) - 2(x + 2)}{(x + 2)(x - 1)} = \frac{3x - 3 - 2x - 4}{(x + 2)(x - 1)} = \frac{x - 7}{(x + 2)(x - 1)}

Leave the denominator factorised unless asked to expand it.

Dividing a quadratic or cubic polynomial by a linear expression can be done using algebraic long division. Follow the cycle: divide the leading term, multiply back, subtract by changing signs, and bring down the next term. Junior Cycle Higher Level divisions always finish with zero remainder. Insert $0x^2$ for a missing $x^2$ term to keep columns aligned.

Changing the Subject of a Formula

Rearranging a formula means isolating the target variable using inverse operations:

Clear fractions by multiplying both sides by the non-zero denominator.
Expand brackets and collect all terms containing the target variable on one side.
Factorise out the target variable, then divide both sides by the remaining non-zero factor.
Undo a square using square roots; include both signs unless the context requires a non-negative value.

Key terms

Variable: A letter that stands for an unknown number or a quantity that can change value.
Coefficient: The numerical factor multiplying a variable, such as the 5 in $5x^2$ .
Like Terms: Terms with identical letters and identical powers that can be added or subtracted together.
Highest Common Factor: The greatest number and variable part that divides into every term of an expression.
Difference of Two Squares: A binomial with two squared terms separated by a minus, factorising as $(a - b)(a + b)$ .
Subject of a Formula: The single variable isolated on one side of an equation without appearing anywhere on the other side.

Check yourself

Expand and simplify $(2x - 3)(x + 5)$ .
$2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15$ .
Factorise $9x^2 - 16y^2$ completely.
$(3x - 4y)(3x + 4y)$ using the difference of two squares.
Find the value of $3x^2 - 5x$ when $x = -2$ .
$3(-2)^2 - 5(-2) = 3(4) + 10 = 12 + 10 = 22$ .
Make $r$ the subject of $V = \pi r^2 h$ , where $r > 0$ and $h \neq 0$ .
Divide both sides by the non-zero factor $\pi h$ to get $r^2 = \frac{V}{\pi h}$ , then take the square root: $r = \sqrt{\frac{V}{\pi h}}$ .

Algebra - Expressions