Algebraic Expressions • Topic 2 of 4

Expanding & the FOIL Method

Expanding turns a product of brackets into a sum of terms. For two binomials, FOIL — First, Outer, Inner, Last — is just the distributive law applied twice: multiply every term in the first bracket by every term in the second. (x + 4)(x + 3) gives x² + 3x + 4x + 12 = x² + 7x + 12. The two middle terms are like terms and almost always combine. Memorise the three special products so you never expand them the long way: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b². The middle-term 2ab in a squared binomial is the value people forget — (x + 5)² is not x² + 25. When a bracket multiplies a trinomial, distribute term by term and stay organised so nothing is skipped.

✅ Solved examples

1. Expand (x + 6)(x − 2).
FOIL: x² − 2x + 6x − 12 = x² + 4x − 12.
2. Expand (2x − 3)(x + 5).
FOIL: 2x·x + 2x·5 − 3·x − 3·5 = 2x² + 10x − 3x − 15 = 2x² + 7x − 15.
3. Expand (3a − 4)².
Use (a − b)² = a² − 2ab + b²: (3a)² − 2(3a)(4) + 4² = 9a² − 24a + 16.
4. Expand (2y + 7)(2y − 7).
Difference of squares: (2y)² − 7² = 4y² − 49.

✏️ Practice — try these, take hints as needed

1. Expand (x + 8)(x + 2).
FOIL.
x² + 2x + 8x + 16.
Combine the middle terms.
x² + 10x + 16
2. Expand (x − 5)(x − 3).
Both signs negative.
x² − 3x − 5x + 15.
Middle = −8x.
x² − 8x + 15
3. Expand (x + 9)².
(a + b)² = a² + 2ab + b².
Do not forget 2ab.
2·x·9 = 18x.
x² + 18x + 81
4. Expand (5 − 2x)(5 + 2x).
Difference of squares.
5² − (2x)².
25 − 4x².
25 − 4x²
5. Expand (x + 1)(x² − 3x + 2).
Distribute x, then 1.
x³ − 3x² + 2x + x² − 3x + 2.
Combine like powers.
x³ − 2x² − x + 2

📝 Topic test — 8 questions

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