Year 10: Expanding algebraic expressions

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Expansion is the process of removing brackets so an expression is written as a sum (or difference) of terms. In Year 10, you’ll expand everything from simple single brackets to binomials like $(a+b)(c–d)$. Mastering this skill sets you up for factorising, simplifying fractions, and tackling quadratic equations in senior maths.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

1. Why do we expand?

Think of brackets as packaging. Sometimes you need the compact form (for factoring or substitution), but often you need everything “unboxed” so like terms are visible. Expansion lets you:

• Combine like terms to simplify expressions.
• Prepare for solving equations or inequalities.
• Find areas (e.g. rectangle with sides $(x+3)$ and $(x–2)$).

2. Single‑bracket expansion

Use the distributive law: $a(b+c)=ab+ac$.

$3(x–4)=3·x–3·4=3x–12–2(5y+7)=–10y–14(\text{watch}\text{the}\text{negative}!)$

Revision of multiplication laws

Exercise

3. Double‑bracket expansion (FOIL)

When two binomials multiply, every term in the first bracket multiplies every term in the second. A handy mnemonic is FOIL—First, Outer, Inner, Last.

$(x+5)(x–2)F:x·x=x\mathrm{²}O:x·(–2)=–2xI:5·x=+5xL:5·(–2)=–10\text{Combine}\text{like}\text{terms}:x\mathrm{²}+3x–10$

Exercise

4. Special products

Recognising patterns speeds you up:

• Square of a sum:$(a+b)\mathrm{²}=a\mathrm{²}+2ab+b\mathrm{²}$
• Square of a difference:$(a–b)\mathrm{²}=a\mathrm{²}–2ab+b\mathrm{²}$
• Difference of squares:$(a+b)(a–b)=a\mathrm{²}–b\mathrm{²}$

$(2x–3)\mathrm{²}=4x\mathrm{²}–12x+9(7y+1)(7y–1)=49y\mathrm{²}–1$

5. Expanding with more than two terms

For $(x+2)(x\mathrm{²}–x+4)$, multiply every term in the first bracket by the entire second bracket, then combine like terms:

$x(x\mathrm{²}–x+4)+2(x\mathrm{²}–x+4)=x\mathrm{³}–x\mathrm{²}+4x+2x\mathrm{²}–2x+8=x\mathrm{³}+x\mathrm{²}+2x+8$

Exercise

6. Common mistakes

• Dropping a term: In double brackets you should end with four products. Count them.
• Sign errors: Remember that a negative times a positive is negative, and negative × negative is positive.
• Forgetting to combine like terms: After expanding, always scan for terms with the same variables and powers.
• Mistaking $(a+b)\mathrm{²}$ for $a\mathrm{²}+b\mathrm{²}$: Don’t skip the middle $2ab$ term.

7. Checking your work

Pick a simple value (e.g. $x=1$) and substitute into both the original and expanded forms. If they match, chances are you’re right.

8. Where expansion pops up

Beyond algebra drills, expansion appears when calculating projectile motion, optimising areas in design tech, and simplifying spreadsheet formulas. Nail it now and you’ll save headaches later.

So next time brackets appear, breathe, distribute methodically, combine like terms, and check with a quick substitution. Expansion is just organised multiplication—once you’ve got the rhythm, it’s smooth sailing!