FOIL Method Calculator – Expand & Factor
FOIL method calculator to expand binomials (ax + b)(cx + d) and factor quadratics ax² + bx + c. Step-by-step First–Outer–Inner–Last working, simplified result, and algebra explanations.
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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
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)
Second binomial (cx + d)
You can also type non-integer values (e.g. 1.5, -0.25). Use dot or comma as decimal separator.
Output is shown as both text and MathJax-formatted formulas.
What is the FOIL method?
The FOIL method is a mnemonic for multiplying two binomials. FOIL stands for:
- First – multiply the first terms in each binomial
- Outer – multiply the outer pair of terms
- Inner – multiply the inner pair of terms
- Last – 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
Formula (LaTeX) + variables + units
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\( (ax + b)(cx + d) = \underbrace{acx^2}_{\text{First}} + \underbrace{adx}_{\text{Outer}} + \underbrace{bcx}_{\text{Inner}} + \underbrace{bd}_{\text{Last}} \) So the simplified form is \( acx^2 + (ad + bc)x + bd \).
\( (ax + b)(cx + d) = ax(cx + d) + b(cx + d) \) \( = ax \cdot cx + ax \cdot d + b \cdot cx + b \cdot d \).
\( (px + q)(rx + s) = prx^2 + (ps + qr)x + qs \) \( pr = a \) \( ps + qr = b \) \( qs = c \)
\( (2x + 3)(x - 5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15 \).
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Last code update: 2026-01-19
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