Once upon a time, two brilliant scientists, dr. See, and Dr. aitch put their brains together to solve mankind's problems. You will never believe what they discovered After years and years of research they invented a gum which, when chewed regularly, in appropriate quantities, would make people smarter. (I told you that you wouldn't believe it) they decided to call it IntelliGum.
   Now, when they marketed IntelliGum they sold it piled in a column. (See the figure below). The number of pieces of IntelliGum in each column depended on the age of the person who was buying it. For example, if a 7 year old bought a column, it must contain 7 pieces of IntelliGum, and if a 20 year old bought a column it must contain 20 pieces of IntelliGum. IntelliGum was also sold in individual packages of one piece, for people who lost some of theirs or who shared some with friends.

Day 2

Packaging

1. A column of IntelliGum can have any number of pieces of gum in it.
   
2. A box of IntelliGum has as many columns of gum as there are pieces of gum in each column.
   
3. A carton of IntelliGum has as many boxes of gum as there are columns of gum in each box.
   
4. A crate of IntelliGum has as many cartons of gum as there are boxes of gum in each carton.

Day 3

Ordering

Last day we discovered that if x represents the number of pieces of IntelliGum in a column, then x² would represent the number of pieces of IntelliGum in a box, x³ would represent the number of pieces of IntelliGum in a carton, and x⁴ would represent the number of pieces of IntelliGum in a crate. A summary follows:

When the IntelliGum company shipped an order, they kept track of the number of pieces of gum sent by using an Algebraic expression called polynomial.

An example would be: 8x⁴ + 4x³ + 7x² + 9x + 20

1. What do you suppose this means?

2. When a wholesaler ordered 5 crates, 6 cartons, 8 boxes, 10 columns, and 13 pieces of IntelliGum, how do you think he would represent the total number of pieces of IntelliGum he had ordered?
      Workers at the truck-loading platform would receive several orders to be delivered in the same truck. They quickly worked out a way to determine how many of each type of container was necessary. Of course they used algebraic expressions all the time.

3. If you were one of the workers and you had to ship the following orders in the same truck, what Algebraic expression could you use to indicate simply, what would be required for the delivery?

      Order #1. 8x⁴ + 7x³ + 9x² + 8x + 80
      Order #2. 2x⁴ + 3x³ + 6x² + 5x + 60

4. Sometimes retailers would compare information about the number of pieces of IntelliGum they had. Write an algebraic expression to show how many more pieces Order # 1 above has compared to Order # 2.

Interpreting Orders

1. If a retailer was missing 4 cartons in order to have 5 complete crates, what algebraic expression could he use to represent the number of pieces of IntelliGum he has.

2. What would a retailer mean if he represented his stock of IntelliGum with the following polynomial? 6x³ - 5x² + 3x - 1

A.  Simplify the following orders

1. (2x³ + 5x² + 4x + 7) + (5x³ + 7x² + 6x + 14)

2. (5x⁴ + 6x³ + 7x² + 9x + 5) + (8x³ +2x + 13)

3. (x⁴ + 2x² + 4x + 1) + (8x³ + 5x² +7)

4. (5x³ - 2x² + 3) + (2x³ + 4x² - 1)

5. (2x⁴ - 7x³ + 4x² - 2x + 9) + (6x⁴ + 2x³ - 5x² - 6x - 3)

6. (3x⁴ - 8x³ + 2x² - 7x + 15) + (2x⁴ + 5x³ - 4x² + x - 7) + (9x³ + 6x² - 8x + 13)

B.  Help the retailers compare their stocks of IntelliGum.

1. (5x² + 8x + 6) - (2x² + 4x + 4)

2. (6x³ - 4x² - 7x + 22) - (5x³ + 7x² - 4x + 22)

3. (5x³ + 6x² - 3x +12) - (3x³ - 7x² + x - 9)

4. (9x⁴ + 5x³ + 7x² + 8x + 24) - (7x⁴ + 3x³ + 4x² + 5x + 16)

5. (8x⁴ + 4x³ - 6x² - 4x + 23) - (2x⁴ - 6x³ + 5x² - 4x + 12)

6. (7x⁴ - 5x³ + 6x² + 8x - 10) - (5x⁴ - 2x³ - 7x² + 12x - 15)