FOIL Method Calculator
Multiply two binomials step-by-step using the FOIL method.
See First, Outer, Inner, and Last terms individually, then the simplified combined result.
The FOIL Method
FOIL is an acronym for multiplying two binomials — expressions with exactly two terms. It stands for: First, Outer, Inner, Last — the four products you compute and then combine.
For (ax + b)(cx + d):
| Step | Product | Result |
|---|---|---|
| First | a·c | coefficient of x² |
| Outer | a·d | coefficient of x (part 1) |
| Inner | b·c | coefficient of x (part 2) |
| Last | b·d | constant term |
Combined result:
(ax + b)(cx + d) = (a·c)x² + (a·d + b·c)x + (b·d)
Example:
(2x + 3)(x − 5)
- First: 2 × 1 = 2 → 2x²
- Outer: 2 × (−5) = −10 → −10x
- Inner: 3 × 1 = 3 → 3x
- Last: 3 × (−5) = −15
Combined: 2x² + (−10 + 3)x + (−15) = 2x² − 7x − 15
Special cases: patterns to recognize:
| Pattern | Identity |
|---|---|
| (a + b)(a − b) | a² − b² (difference of squares) |
| (a + b)² | a² + 2ab + b² (perfect square) |
| (a − b)² | a² − 2ab + b² (perfect square) |
Why FOIL matters:
FOIL is the foundation of polynomial multiplication. Understanding it makes factoring (the reverse operation) much easier. Factoring is essential for solving quadratic equations and simplifying rational expressions.
Entering negative coefficients:
Use negative values for b or d to represent subtraction. For (3x − 4), enter a = 3 and b = −4.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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