1. P.3 is about Polynomials What’s a polynomial? How to combine like terms? How to multiplying polynomials? What are special products?

2. 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.

3. 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

4. Subtracting Polynomials One 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.

5. (7x4-x2-4x+2) + -1(3x4-4x2+3x) = (7x4-x2-4x+2) + (-3x4+4x2-3x) NOW associativity/commutativity apply:

6. 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.

7. 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)

8. (5x+3)(-6x2+15x-4) For each arrow, you’ll have a new term:

9. Remember this rule works for all of them: “Each term of 1stpolynomial times each term of 2ndpolynomial.” You probably learned FOIL (First Outer Inner Last) in high school, which only works for binomials x binomials. (2x – 4)(x + 5)

10. 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 NONO.

11. Freshman Dream I guarantee I’ll try to trick you on exams. Don’t ever ever fall for this.

12. Patterns (u + v)(u – v) = (u + v)2 = Use this as a “shortcut” for this one: (3x + 2)2

13. 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: