9 - 1 2 . A . A P R . 1 U N D E R STA N D T H AT P O LY N O M I A L S F O R M A S Y ST E M A N A L O G O U S

T O T H E I N T E G E R S, N A M E LY, T H E Y A R E C L O SE D U N D E R T H E O P E R AT I O N S O F

A D D I T I O N, S U B T R A C T I O N, A N D M U LT I P L I C AT I O N; A D D, S U B T R A C T, A N D

M U LT I P LY P O LY N O M I A L S.

POLYNOMIALS

What are polynomials and how do they work?

Polynomial Definition:

A polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate (or variable), x, is , which is a quadratic polynomial.

Using the same polynomial above , the variable (indeterminate) is x and the coefficients are 1, -4, and 7. What are our exponents? How many are there?

What are the types of polynomials?

“What makes a polynomial a function, and what makes a function a polynomial?”

First of all, all polynomials are functions because they take inputs of whatever the domain of your independent variable is (most often x) and maps it to a dependent variable (most often y). Even when we no longer deal with x’s and y’s, the concept stays the same in whatever dimension you choose.

It’s important to note that not all functions are polynomials. Only functions in this form are polynomial functions…

What can we do with polynomials?

Polynomials can be used similar to how we use number integers. This means that they are closed under addition, subtraction, and multiplication. This means when we add, subtract, or multiply two polynomials, we get another polynomial as our result.

Why do we run into trouble when we divide?

Adding Polynomials

Adding polynomials is like adding any other number, but we have to be sure we’re adding the same thing. You can only add coefficients that are multiplied by the same power of x.

Let’s look at an example

Subtracting Polynomials

This is the same as just adding the negation of a polynomial. Meaning we can multiply the polynomial we’re subtracting by -1 and then add the result to the other polynomial.

Multiplying Polynomials

We will be focusing the most on multiplying since this is more difficult. Recall foiling. (x-1)(x+2) … It’s the same idea, except now we have more terms.

We want to make sure we multiply every term on the left by every term on the right.

Examples

Dividing polynomials

Polynomials are not closed under division, meaning the result may not be a polynomial… But, we can still divide them, and we will!

Notice we can get a remainder. What do you think we’ll do

with the remainder?

What to do with the remainder

When we have a remainder after dividing, we put it over what we were dividing by and add it to the rest of our answer. Example…