Polynomials • Topic 3 of 4

Multiplication of Polynomials

To multiply polynomials, multiply every term of the first by every term of the second, then combine like terms. For two binomials this is FOIL (First, Outer, Inner, Last): (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6. Two special products save time: the difference of squares (a + b)(a − b) = a² − b², and the perfect square (a + b)² = a² + 2ab + b². Distributing carefully and tracking signs prevents errors. Polynomial multiplication is essential for expanding expressions and recognising patterns the SAT tests in quadratics.

✅ Solved examples

1. Expand (x + 3)(x + 2).
FOIL: x² + 2x + 3x + 6 = x² + 5x + 6.
2. Expand (x + 5)(x − 5).
Difference of squares: x² − 25.
3. Expand (x + 4)².
Perfect square: x² + 2(4)x + 16 = x² + 8x + 16.
4. Expand 2x(x + 3).
Distribute: 2x² + 6x.

✏️ Practice — try these, take hints as needed

1. Expand (x + 1)(x + 6).
Use FOIL.
x² + 6x + x + 6.
Combine.
x² + 7x + 6.
2. Expand (x + 7)(x − 7).
Difference of squares.
x² − 7².
—
x² − 49.
3. Expand (x − 3)².
Perfect square (a − b)².
x² − 2(3)x + 9.
—
x² − 6x + 9.
4. Expand (2x + 1)(x + 4).
FOIL.
2x² + 8x + x + 4.
Combine.
2x² + 9x + 4.
5. Expand 3x(2x − 5).
Distribute 3x.
3x·2x and 3x·(−5).
—
6x² − 15x.

📝 Topic test — 8 questions

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