Foil

Your Friend:

F.O.I.L

A polynomial is an expression made with constants, variables and exponents, which are combined using addition, subtraction and multiplication, ... but not division.

Terms are made up of coefficients and/or variables.

x, 4, -2x, 3x2

Note: A polynomial expression with one term is called a monomial.

Polynomial expressions with two terms are called binomials.

3x2 – 4x4, -11x7 + 9x13

We know how to combine like terms, and we know properties of exponents . . .So how would we expand a monomial times a binomial?

3x3 (4x2 – 5x)

!3x3 (4x2 – 5x)

^ This monomial needs to be DISTRIBUTED to both terms within the binomial . . .

hmmm….

Distributive Property:a(b + c) = a(b) + a(c)a(b - c) = a(b) – a(c)

So, 3x3 (4x2 – 5x) = 3x3 (4x2) – 3x3(5x)

= 12x5 – 15x4

Multiplying a binomial times a binomial also uses the Distributive Property.

(x + 11)(x – 4)

Let m = (x + 11) n = x p = 4

Then we have m(n - p)…Use the Distributive Property!m(n - p) = m(n) – m(p)Now substitute the original values back in…(x + 11)(x) – (x + 11)(4)

The Distributive Property can be used again on both pieces,(x2 + 11x) – (4x + 44)Now, drop the parentheses and combine like terms:x2 + 11x – 4x – 44 = x2 + 7x - 44

We can shortcut some of the previous steps by using a helpful acronym.

F.O.I.L

(This acronym is NOT its own property, but a derivation of the Distributive Property!)

irst

uter

nner

ast

When multiplying two binomials, F.O.I.L reminds us how to multiply the terms within the binomials!

(x + 11)(x – 4) =

Let’s try the previous problem, this time remembering F.O.I.L(x + 11)(x – 4)We do the steps of F.O.I.L in order:

“First” reminds us to multiply the first term in each binomial together.(x + 11)(x – 4)= x2

“Outer” reminds us to multiply the outermost terms together.(x + 11)(x – 4)= -4x“Inner” refers to multiplying the innermost terms.(x + 11)(x – 4)= 11x“Last” reminds us to multiply the last term in each binomial together.(x + 11)(x – 4)= -44

(x + 11)(x – 4) = x2 (x + 11)(x – 4) = x2 - 4x (x + 11)(x – 4) = x2 - 4x + 11x (x + 11)(x – 4) = x2 - 4x + 11x – 44

That was much easier!

Remember, F.O.I.L can be used when multiplying any binomial by another binomial.