How does the FOIL method work?
The FOIL method gives you a fixed order for the four products you get when two binomials multiply. Each letter names one pair of terms.
| Letter | Pair | In (a + b)(c + d) |
|---|---|---|
| F | First terms of each bracket | a × c |
| O | Outer terms | a × d |
| I | Inner terms | b × c |
| L | Last terms of each bracket | b × d |
The outer and inner products are usually like terms, so they combine into the middle term of the answer. That is why two binomials normally give three terms.
Worked example: (2x − 3)(x + 4)
The answer is 2x² + 5x − 12. Keep the minus sign with the 3 in every product it takes part in.
Check with x = 1: (2 − 3)(1 + 4) = −5, and 2 + 5 − 12 = −5. The values agree, so the expansion is right.
What happens when you FOIL (x + 5)(x − 5)?
You get x² − 25, because the outer and inner products cancel. This special product is called a difference of squares.
First gives x², Outer gives −5x, Inner gives +5x, and Last gives −25. The −5x and +5x add to zero. Any pair like (a + b)(a − b) gives a² − b², so (x − 3)(x + 3) is x² − 9.
Compare that with (x + 2)(x − 3), where the numbers differ. The outer and inner products are −3x and 2x, which only partly cancel, so the answer is x² − x − 6. The middle term vanishes only when the two numbers match.
How do you FOIL a squared binomial like (x − 4)²?
Write it as (x − 4)(x − 4) and FOIL as usual. The result is x² − 8x + 16.
The pattern is (a − b)² = a² − 2ab + b², and (a + b)² = a² + 2ab + b². The middle term is twice the product, and the last term is always positive. Writing (x − 4)² as x² − 16 is the classic mistake here.
Why does FOIL only work for two binomials?
FOIL names exactly four products, and only a pair of two-term brackets produces four. A binomial times a trinomial produces six products, so FOIL would leave two out.
For bigger products, use the box method or distribute each term. In the box method you draw a grid with one polynomial along the top and the other down the side, then fill each cell with a product. Distributing works the same way without the grid: every term in the first bracket multiplies every term in the second.
For example, (x − 2)(x² + 2x + 4) has six products and simplifies to x³ − 8. The expand calculator handles these longer products.
How does the box method compare with FOIL?
The box method gives the same four products as FOIL, laid out in a grid. Some students find it easier because every product has its own cell, so nothing gets skipped.
For (2x − 3)(x + 4), write 2x and −3 down the side and x and 4 across the top. Each cell is the product of its row and column.
| × | x | 4 |
|---|---|---|
| 2x | 2x² | 8x |
| −3 | −3x | −12 |
Add all four cells: 2x² + 8x − 3x − 12 = 2x² + 5x − 12. The like terms sit on a diagonal of the grid, which makes them easy to find. The grid just grows a row or column when a factor has more terms, which is why it keeps working where FOIL stops.
How do you check a FOIL answer?
Substitute a small number into both the original and your answer and compare. If the two values match, your expansion is very likely right.
Try (3x + 1)(2x + 5), which expands to 6x² + 17x + 5. At x = 1 the original is 4 × 7 = 28, and 6 + 17 + 5 = 28. Avoid testing with x = 0 alone, since it only checks the constant term.
What mistakes do students make with FOIL?
Most mistakes are sign errors in the Inner and Last steps. In (2x − 3)(x + 4), the inner product is −3x, not 3x. Circle the sign with each term before you start.
Another is forgetting to combine the Outer and Inner products at the end, leaving four terms where three belong. And with coefficients, multiply them fully: in (3x + 1)(2x + 5), First is 6x², not 5x² or 6x.