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Home › Math & Conversions › Core Math & Algebra › FOIL Method Calculator FOIL Method Calculator – Expand & Factor This FOIL method calculator helps you expand products of binomials \((ax + b)(cx + d)\) and factor quadratics \(ax^2 + bx + c\) using the reverse FOIL approach. It shows every step – First, Outer, Inner, Last – and gives the simplified polynomial, so you can check homework, build intuition, or prepare lesson material. FOIL method calculator Mode Expand (ax + b)(cx + d) Factor ax² + bx + c (reverse FOIL) Variable symbol Usually x, y, t… Single letter recommended. Enter the coefficients of your binomials in the form (a·x + b)(c·x + d) . Coefficients can be negative or decimal. First binomial (ax + b) a (coefficient of variable) b (constant term) Second binomial (cx + d) c (coefficient of variable) d (constant term) You can also type non-integer values (e.g. 1.5, -0.25). Use dot or comma as decimal separator. Variable symbol Usually x, y, t… Single letter recommended. Enter coefficients of your quadratic \(ax^2 + bx + c\). The tool searches for an integer factorization of the form \((p x + q)(r x + s)\). a (coefficient of x²) b (coefficient of x) c (constant term) If no integer factorization is found, the calculator tells you and suggests using the quadratic formula. Calculate Clear Output is shown as both text and MathJax-formatted formulas. Result Step-by-step What is the FOIL method? The FOIL method is a mnemonic for multiplying two binomials. FOIL stands for: F irst – multiply the first terms in each binomial O uter – multiply the outer pair of terms I nner – multiply the inner pair of terms L ast – multiply the last terms in each binomial For binomials \((ax + b)\) and \((cx + d)\), FOIL gives: \( (ax + b)(cx + d) = \underbrace{acx^2}_{\text{First}} + \underbrace{adx}_{\text{Outer}} + \underbrace{bcx}_{\text{Inner}} + \underbrace{bd}_{\text{Last}} \) So the simplified form is \( acx^2 + (ad + bc)x + bd \). FOIL is just the distributive property Conceptually, FOIL is nothing more than applying the distributive property twice: \( (ax + b)(cx + d) = ax(cx + d) + b(cx + d) \) \( = ax \cdot cx + ax \cdot d + b \cdot cx + b \cdot d \). FOIL is handy for binomials, but for larger expressions (like trinomials) it is clearer to think in terms of full distribution. Reverse FOIL: factoring quadratics Factoring a quadratic polynomial \( ax^2 + bx + c \) into the product of two binomials \((px + q)(rx + s)\) is sometimes called reverse FOIL : \( (px + q)(rx + s) = prx^2 + (ps + qr)x + qs \) \( pr = a \) \( ps + qr = b \) \( qs = c \) To factor \( ax^2 + bx + c \) using integer reverse FOIL, you look for integers \( p, q, r, s \) that satisfy those three relationships. The calculator automates that search and either returns a neat factorization or tells you that no integer factorization exists. FOIL method – worked example Expand \((2x + 3)(x - 5)\). First: \( 2x \cdot x = 2x^2 \) Outer: \( 2x \cdot (-5) = -10x \) Inner: \( 3 \cdot x = 3x \) Last: \( 3 \cdot (-5) = -15 \) Now add all terms and combine like terms: \( (2x + 3)(x - 5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15 \). FOIL method – frequently asked questions Can I use FOIL for (x + 1)(x² + 2x + 3)? + No – FOIL is specifically for multiplying two binomials . For \((x + 1)(x^2 + 2x + 3)\) you should use the distributive property: \( (x + 1)(x^2 + 2x + 3) = x(x^2 + 2x + 3) + 1(x^2 + 2x + 3) \), then expand each part and combine like terms. The idea is the same, but FOIL as a mnemonic doesn’t cover trinomials directly. What if some coefficients are fractions? + FOIL works with any real numbers, including fractions and decimals. Just multiply normally and simplify at the end. This calculator accepts decimal input (for example 0.5 or -1.25) and returns a simplified decimal result. Does the order of the binomials matter? + No. Multiplication is commutative, so \((ax + b)(cx + d)\) and \((cx + d)(ax + b)\) expand to the same result. You may get the terms in a different intermediate order, but the simplified polynomial is identical. How is reverse FOIL related to the quadratic formula? + Reverse FOIL is a structured guess-and-check method to factor quadratics with “nice” integer roots. The quadratic formula works for all quadratics, and it will find roots even when no integer factorization exists. If reverse FOIL fails but you still need the exact roots, use the quadratic formula instead. Core Math & Algebra tools Basic Arithmetic Standard Form Calculator Percent Error Calculator Effect Size (Cohen’s d) G-test (Likelihood-Ratio Test) FOIL Method Calculator – you are here Graph Adjacency Matrix Speed, Distance, Time Friedman Test More algebra & polynomial tools Quadratic Equation Solver Polynomial Long Division Polynomial Factorization Linear Equation Solver System of Equations Solver Exponent & Radical Calculator
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