FOIL Method Calculator
Expand the product (ax+b)(cx+d) using the FOIL method with step-by-step working. Shows First, Outer, Inner, and Last terms separately.
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How to use this calculator
FOIL stands for First, Outer, Inner, Last — the four pairs of terms multiplied when expanding the product of two binomials.
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Enter a and b for the first binomial (ax + b).
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Enter c and d for the second binomial (cx + d).
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The expanded polynomial is shown along with each FOIL step labelled.
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To use negative terms, enter negative values for b or d (e.g., b = −3 gives x − 3).
Frequently asked questions
What does FOIL stand for?
FOIL stands for First, Outer, Inner, Last — the order in which you multiply pairs of terms from the two binomials: First (a×c), Outer (a×d), Inner (b×c), Last (b×d). Adding all four products and combining like terms gives the expanded polynomial.
Does FOIL work for more than two terms?
FOIL specifically applies to two binomials (two-term expressions). For trinomials or longer polynomials, use the distributive property systematically: multiply each term of the first polynomial by each term of the second, then collect like terms. The number of partial products is (terms in first) × (terms in second).
How do I verify the FOIL expansion?
Substitute a specific value for x into both the factored form (ax+b)(cx+d) and the expanded form, and check they give the same result. For example, with x = 1: (2·1+3)(4·1−1) = 5×3 = 15; expanded: 8−2+12−3 = 15. ✓
What is the reverse of FOIL (factoring)?
Factoring a quadratic ax²+bx+c finds two binomials whose product equals the quadratic. For monic quadratics (a=1), find two numbers that multiply to c and add to b: x²+5x+6 = (x+2)(x+3) because 2×3=6 and 2+3=5. For non-monic quadratics, use the AC method or the quadratic formula.
FOIL calculator — expand binomial products step by step
FOIL as the distributive property
FOIL is a mnemonic for applying the distributive property twice. (ax+b)(cx+d) = ax(cx+d) + b(cx+d) = acx² + adx + bcx + bd. Grouping the x terms gives acx² + (ad+bc)x + bd. The FOIL acronym simply helps students remember the four pairs: First (the x² term), Outer and Inner (the x terms), and Last (the constant).
Special binomial products worth memorising
Three patterns arise so often they deserve memorisation: (x+a)² = x²+2ax+a² (perfect square trinomial); (x−a)² = x²−2ax+a² (perfect square trinomial); (x+a)(x−a) = x²−a² (difference of squares). These templates let you skip FOIL for common cases. For example, (x+5)(x−5) = x²−25 immediately — no multiplication needed.
FOIL in algebra and polynomial arithmetic
Polynomial multiplication underlies many advanced topics: multiplying complex numbers (treated as binomials in the imaginary unit i), computing convolutions in signal processing, multiplying polynomials in abstract algebra and cryptography (polynomial rings over finite fields). The step-by-step FOIL process, scaled up, is the algorithm behind all polynomial multiplication and is the foundation of the schoolbook multiplication algorithm you learned for integers.
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