1) A polynomial is an algebraic expression involving variables, constants, and exponents combined using addition, subtraction, and multiplication but not division. 2) Terms are made up of coefficients and/or variables such as x, 4, -2x, 3x^2. 3) The distributive property, a(b + c) = ab + ac, can be used to expand a monomial multiplied by a binomial such as 3x^3(4x^2 - 5x) = 12x^5 - 15x^4.

1. Your Friend: F.O.I.L

2. 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.

3. 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)

4. 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

5. 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

6. We can shortcut some of the previous steps by using a helpful acronym. (This acronym is NOT its own property, but a derivation of the Distributive Property!) F.O.I.L irst uter nner ast When multiplying two binomials, F.O.I.L reminds us how to multiply the terms within the binomials!

7. 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 - 4x + 11x – 44

8. That was much easier! Remember, F.O.I.L can be used when multiplying any binomial by another binomial.