Multiplying Binomials (FOIL)

When you multiply polynomials, you must make sure that you multiply each term by each other term.  When multiplying binomials, we ensure that each term is multiplied by each other term using the FOIL process.

Foil is an acronym that stands for first, outer, inner, last.

The basic process of using foil, is to take the two binomials and first multiply the "first" terms of each binomial together.

$\require{cancel}$

$\eqalign{(\color{red}x\color{black}+3)(\color{red}x\color{black}-5)\\\text{First:} \quad x \times x =\color{red}{ x^2}\\(x+3)(x-5)=\color{red}{x^2}\text{...}}$

Then multiply the "outer" terms of each binomial together.

$\eqalign{(\color{blue}{x}\color{black}+3)(x\color{blue}{-5}\color{black})\\\text{Outer:} \quad x \times -5 = \color{blue}{-5x}\\(x+3)(x-5)=\color{red}{x^2}\color{blue}{-5x}\text{...}}$

Then multiply the "Inner" terms of each binomial together.

$\eqalign{(x\color{green}{+3}\color{black})(\color{green}{x}\color{black}-5)\\\text{Inner:} \quad 3 \times x = \color{green}{3x}\\(x+3)(x-5)=\color{red}{x^2} \color{blue}{-5x}\color{green}{+3x} \text{...}}$

Finally, multiply the "Last" terms of each binomial together.  That will result in 4 terms, but the middle two terms will be like terms, so you will combine them together to create a trinomial (three terms).

$\eqalign{(x\color{purple}{+3}\color{black})(x\color{purple}{-5}\color{black})\\\text{Last:} \quad 3 \times -5 =\color{purple}{ -15}\\(x+3)(x-5)=\color{red}{x^2}\color{blue}{-5x}\color{green}+3x\color{purple}{-15}\\\text{Combine like terms:}\\x^2-2x-15}$

You can use the FOIL process to multiply any two binomials.  

When you are working with a quadratic equation, the "inner" and "outer" terms are typically like terms that need to be combined, resulting in a final trinomial.