What does FOIL stand for in algebra?

FOIL is a mnemonic for multiplying two binomials. It stands for First, Outer, Inner, Last: multiply the first terms, then the outer terms, the inner terms, and finally the last terms, and then combine like terms.

When can I use the FOIL method?

You can use FOIL whenever you multiply two binomials of the form (ax + b)(cx + d). For products with more terms, such as trinomials, it is better to use the distributive property systematically instead of FOIL.

What is the general formula for (ax + b)(cx + d)?

The product of two binomials (ax + b)(cx + d) is ax·cx + ax·d + b·cx + b·d = acx² + (ad + bc)x + bd. The FOIL method is a structured way to remember these four partial products.

Can FOIL be used to factor quadratics?

Yes. Factoring a quadratic ax² + bx + c into (px + q)(rx + s) is sometimes called the reverse FOIL method: you look for numbers p, q, r, s that multiply and add to the right coefficients. This works neatly when integer factors exist.

What if my quadratic does not factor nicely?

If a quadratic ax² + bx + c has no integer factorization, you can still solve it using the quadratic formula or by completing the square. In that case, reverse FOIL over the integers will not succeed, even though the quadratic may have real or complex roots.

FOIL Method Calculator – Expand & Factor Binomials

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.

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 Calculator – Expand & Factor