Expand binomial products (ax+b)(cx+d) using the FOIL method. First, Outer, Inner, Last—perfect for mastering polynomial expansion.
Last updated: May 2026 | By Patchworkr Team
| Binomials | Expanded |
|---|---|
| (x + 2)(x + 3) | x² + 5x + 6 |
| (2x - 1)(x + 4) | 2x² + 7x - 4 |
| (x - 3)(x - 5) | x² - 8x + 15 |
| (3x + 2)(2x + 5) | 6x² + 19x + 10 |
| (x + 1)(x + 1) | x² + 2x + 1 |
FOIL is a mnemonic for multiplying two binomials: First, Outer, Inner, Last. It stands for the four multiplication steps needed to expand $(ax+b)(cx+d)$ correctly. This method prevents missing or duplicating terms and creates a memorable structure for mental calculation.
FOIL is fundamental in algebra and appears everywhere: solving quadratic equations, graphing parabolas, polynomial manipulation, and modeling phenomena. Understanding FOIL deeply builds intuition for the distributive property and prepares you for binomial theorem, partial fractions, and polynomial division.
For (ax + b)(cx + d):
It's a mnemonic: First, Outer, Inner, Last—the order you multiply binomial terms.
Yes, for any two binomials of any degree or complexity, though the notation may change.
FOIL still works—those terms simply become zero and vanish in the final result.
Apply FOIL to two, then FOIL the result with the third. Repeat as needed.
FOIL is a special application of the distributive property specifically for binomials.
When multiplying trinomials or higher, use the distributive property directly or long multiplication.
Factoring is the reverse: find binomials whose FOIL expansion gives your polynomial.
Yes: $(ax+b)² = a²x² + 2abx + b²$. This is a special FOIL case.
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