# Polynomials

I have a request to go over polynomials, and to specifically explain what FOIL means.

"Polynomials" refer to mathematical expressions that contain multiple terms (poly = multiple, nomials = terms/numbers (roughly)). "Monomial" refers to a polynomial that has only a single (mono) term. Furthermore, the product of multiplying monomials together also results in a monomial. Basically, think of it along these lines:

- 1 term (bunch of things multiplied/divided/raised to a power) = monomial

- more than 1 term (monomials) that are added or substracted = polynomial

Some examples hopefully clear this up:

1) Monomials:
x,
2x,
x^3,
xy,
5xy,
(10xy^4)/3

2) Polynomials:
x+5,
2x-5,
(5xyz^3)/4 + 7x

Doing calculations with polynomials is fairly straight-forward, but you do have to keep a few things in mind.

For addition or subtraction of polynomials, you must remember that you can only combine 'like terms.' 'Like terms' contain the sample variables and differ by the numeric coefficient in the front. 3x and 5x are like terms, and can be added to get 8x. However, 3x and x^2 are not like terms and CANNOT be added like in the first example.

x + 2x = 3x
4x + 4x = 8x
15x + 100x = 115x
50x - 10x = 40x

You cannot add terms in this way that have different variables:
x^2 + 5x
3x^4 - 2x^3
x + 7

**Technically, some expressions such as these can be SIMPLIFIED by pulling out a common factor, but they still cannot be added up, as with like terms:
eg. x^2 + 5x = x(x + 5)

Multiplying polynomials is a little more complicated, but still straight-forward, provided that you keep track of what you are doing. When multiplying two polynomials together, all that you need to do is add up the products of multiplying each term in the first polynomial by each term in the second polynomial. You have probably heard the expression "FOIL" when talking about multiplying polynomials. FOIL is short for "First, Outside, Inside, Last" and refers to which terms you multiply together and add up when multiplying two polynomials, each composed of two monomials (specifically, this is the product of two BINOMIALS).

Here is an example of this:

(2x + 3)(x + 5) = ?

First: 2x * x
Outside: 2x * 5
Inside: 3 * x
Last: 3 * 5

Adding these up gives:
2x^2 + 10x + 3x + 15

Furthermore, as I showed earlier, about adding polynomials with like terms, you can simplify this expression:
2x^2 + 13x + 15

And that's the final, reduced answer. You can apply the FOIL principle to any two binomials to arrive at their product. When you have more complicated polynomials, such as those composed of 3, 4, 5, or more monomials, you do the same type of thing... what I find easiest is to take the first term of the first polynomial, and multiply it with every term of the second polynomial. Then do the same for the second term in the first one, multiplying with every term in the second one, and so on.

Like this:
(x^2 + x + 5)(x^3 + x^2 + 1)

First group (multiply x^2 with all in the second polynomial)
x^2 * x^3
x^2 * x^2
x^2 * 1

Second group (x)
x * x^3
x * x^2
x * 1

Third group (5)
5 * x^3
5 * x^2
5 * 1

Now, you just add up all these terms, and simplify where you can:
(x^2 * x^3) + (x^2 * x^2) + (x^2 * 1) + (x * x^3) + (x * x^2) + (x * 1) + (5 * x^3) + (5 * x^2) + (5 * 1)

(x^5) + (x^4) + (x^2) + (x^4) + (x^3) + (x) + (5x^3) + (5x^2) + (5)

(x^5) + 2(x^4) + 6(x^3) + 6(x^2) + x + 5

And that's it. A little more complicated, but as long as you keep track of what you're doing and work your way through it, you will arrive at the answer!