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The FOIL method is a mnemonic for multiplying two binomials: multiply the First terms, then the Outer terms, then the Inner terms, then the Last terms, and sum all four products. For (ax + b)(cx + d): First = ac·x², Outer = ad·x, Inner = bc·x, Last = bd.
This calculator takes any two binomials of the form (ax + b)(cx + d), applies the FOIL method step by step, combines like terms, and shows the simplified polynomial result. Each of the four FOIL products is displayed separately before combining.
FOIL is actually a special case of the distributive property applied twice: (a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd. FOIL just names the four products to help you remember all four. For trinomials and higher-degree polynomials, FOIL does not apply directly — you must use the full distributive property, multiplying every term in the first polynomial by every term in the second.
Key special products that follow the FOIL pattern and should be memorized: (a + b)² = a² + 2ab + b² (perfect square trinomial), (a − b)² = a² − 2ab + b² (perfect square), (a + b)(a − b) = a² − b² (difference of squares). Recognizing these patterns allows you to skip the FOIL steps entirely and write the answer directly — a major time-saver in algebra and factoring.
(ax+b)(cx+d) = ac·x² + (ad+bc)·x + bd | F=ac·x², O=ad·x, I=bc·x, L=bd
Expand (ax + b)(cx + d)
| Pattern Name | Factored Form | Expanded Form | Key Feature |
|---|---|---|---|
| Perfect square (sum) | (a + b)² | a² + 2ab + b² | Middle term = 2ab |
| Perfect square (diff) | (a − b)² | a² − 2ab + b² | Middle term = −2ab |
| Difference of squares | (a + b)(a − b) | a² − b² | Middle terms cancel |
| Sum × general | (x + p)(x + q) | x² + (p+q)x + pq | Reverse FOIL factoring |
| Double coefficient | (2x + 3)(x − 5) | 2x² − 7x − 15 | a≠1 requires care |
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.