P.3 is about Polynomials

What’s a polynomial?How to combine like terms?

How to multiplying polynomials?What are special products?

What’s a polynomial?

Here’s an example of one

(in descending order, or “standard form”):

2x3-5x2+1

Degree = highest power of x.

Leading Coefficient = coefficient of the term with the highest power of x.

Constant Term = the term with no variables.

Combining Like Terms

A polynomial plus another polynomial would look like:

(5x3-7x2-3) + (x3+2x2-x+8)

By associativity of addition, these parentheses aren’t necessary.

By commutativity, we can rearrange the terms also:

5x3 + x3 – 7x2 + 2x2 – x – 3 + 8

Subtracting PolynomialsOne polynomial minus another polynomial would

look like:(7x4-x2-4x+2) – (3x4-4x2+3x)

CAUTION! This time the ( ) don’t vanish. (No associativity for subtraction) But since minus is same as plus a negative,(7x4-x2-4x+2) + -(3x4-4x2+3x)Now you distribute the -1.

(7x4-x2-4x+2) + -1(3x4-4x2+3x)

= (7x4-x2-4x+2) + (-3x4+4x2-3x)

NOW associativity/commutativity apply:

Summary of add/subtracting

Adding polynomials: the ( ) go away and combine like terms.

Subtracting polynomials: DISTRIBUTE A NEGATIVE ONE and then ( ) go away so you can combine like terms.

Multiplying Polynomials

A polynomial times another looks like:

(5x+3)(-6x2+15x-4)

Each term in the first polynomial must be multiplied by each term in the second polynomial. Draw these arrows!

(5x+3)(-6x2+15x-4)

(5x+3)(-6x2+15x-4)

For each arrow, you’ll have a new term:

Remember this rule works for all of them: “Each term of 1st polynomial times each term of

2nd polynomial.”You probably learned FOIL (First Outer Inner

Last) in high school, which only works for binomials x binomials.

(2x – 4)(x + 5)

Special Products

There are some patterns you can recognize.

Try multiplying out (5x + 9)(5x – 9)

But will this work for (5x + 9)2, which is the same as (5x + 9)(5x + 9)? NO NO NO.

Freshman Dream

I guarantee I’ll try to trick you on exams.

Don’t ever ever fall for this.

Patterns

(u + v)(u – v) =

(u + v)2 =

Use this as a “shortcut” for this one:

(3x + 2)2

MEMORIZE this formula for a profit function

Profit = Revenue – Cost

Let’s say the revenue function is R(x)=45x and the cost function is C(x)=25x+10000. Find the profit function: