Algebraic Expressions

An expression is a mix of letters and numbers joined by $+$ and $-$ signs, with no equals sign. Each separate part is a term, and the number multiplying the letter in a term is the coefficient.

At Stage 8 you meet terms with powers, so reading carefully matters more than ever. In $5x^2 - 3x + 8$ there are three terms: $5x^2$, $-3x$ and $8$. The coefficient of the first term is 5, of the second is $-3$, and $8$ is a constant term.

The conventions of algebra keep everything tidy: hide the times sign ($3 \times n$ becomes $3n$), put the number first ($5y$, never $y5$), write $x$ for $1x$, write $\tfrac{n}{4}$ for $n \div 4$, and use a power for a letter times itself ($n \times n = n^2$).

Like terms have exactly the same letter part. To simplify an expression you collect like terms, adding or subtracting their coefficients.

At Stage 8 the catch is powers. $3x$ and $5x$ are like terms, but $3x$ and $5x^2$ are not — the letter parts differ. So $3x + 5x^2$ cannot be made any shorter.

So $4x^2 + 3x + 2x^2 - x$ simplifies to $6x^2 + 2x$. Watch the signs: a lone $-x$ has a coefficient of $-1$, so $3x - x = 2x$.

The same idea handles two-letter terms: $5ab$ and $2ab$ are like terms, but $5ab$ and $5a$ are not. Always match the whole letter part before you combine.

When you multiply two terms together, you handle the numbers and the letters separately.

To work out $3a \times 4b$: multiply the numbers $3 \times 4 = 12$, then multiply the letters $a \times b = ab$. The result is $12ab$.

When the same letter appears in both terms, multiplying it by itself makes a power. Remember $x \times x = x^2$. So:

• $2x \times 5x = 10x^2$ — the numbers give 10, and $x \times x = x^2$.
• $4n \times n = 4n^2$.
• $a \times a \times a = a^3$.

This is the first law of indices: when you multiply powers of the same letter, you add the indices. You will meet it formally soon, but the picture — letters stacking up — already makes sense.

Dividing terms is easiest if you write the division as a fraction, then simplify it just like a number fraction.

Take $10xy \div 2x$, written as $\tfrac{10xy}{2x}$. Divide the numbers: $10 \div 2 = 5$. Then cancel the matching letter: there is an $x$ on the top and an $x$ on the bottom, so they cancel. The $y$ has nothing to cancel with, so it stays.

So $\tfrac{10xy}{2x} = 5y$. A few more:

• $\tfrac{12a}{3} = 4a$ — only the numbers divide; the $a$ stays.
• $\tfrac{6x^2}{2x} = 3x$ — one $x$ cancels, leaving a single $x$.
• $\tfrac{8ab}{8ab} = 1$ — everything cancels.

If a letter on the bottom has no match on top, it stays on the bottom as a fraction.

To expand a bracket you multiply every term inside it by the term outside. At Stage 8 the term outside is often a letter or a negative number, so the multiplying skills from earlier come straight in.

So $x(x + 3) = x \times x + x \times 3 = x^2 + 3x$.

Watch the signs carefully:

• $5(2y - 3) = 10y - 15$.
• $2x(3x + 4) = 6x^2 + 8x$.
• $-3(x + 2) = -3x - 6$ — a negative outside flips both signs.

After expanding, you often collect like terms: $3(x + 2) + 2(x + 1) = 3x + 6 + 2x + 2 = 5x + 8$.

Sharpening your expression skills now opens up the rest of Stage 8 algebra and beyond:

• Expansion and factorisation reverses what you have just learned — factorising pulls a common factor back out.
• Equations and inequalities are solved much faster when you can simplify both sides first.
• Substitution and formulae rely on reading $3n$ and $x^2$ correctly before you put numbers in.
• Indices grow directly out of multiplying and dividing terms of the same letter.

In everyday life, expressions model patterns and pricing — like the cost of $n$ items at $8 each plus a $3 fee, written $8n + 3$, or the area of a rectangle $x$ by $x + 3$, written $x^2 + 3x$.

If algebra ever feels hard later, it is often a like-terms slip or a sign error underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.