Math I, Sections 2.1 – 2.4 Standard: MM1A2c Add, subtract, multiply and divide polynomials. Today’s Question: What are polynomials, and how do we add, subtract and multiply them? Standard: MM1A2c.

 A monomial is a number, variable, or the product of a number and one or more variables with whole number exponents.  Whole numbers are 0, 1, 2, 3, …  The following are not monomials: xx -1 x 1/2 2 x 1log x ln x yx 2 sin xcos xtan x

Monomial, Binomial, Trinomial - # of terms Degree – add the exponents of each variable within each term. The term with the highest sum defines the degree of the expression. Make a graphic organizer showing the possibilities Degree # Terms ZeroFirstSecondThird Monomial Binomial Trinomial

Monomial, Binomial, Trinomial - # of terms Degree – add the exponents of each variable within each term. The term with the highest sum defines the degree of the expression. Make a graphic organizer showing the possibilities Degree # Terms ZeroFirstSecondThird Monomial25x, 3y, 4z3x 2, 4xy6x 3, xy 2 Binomial2 + 52x + 4, 3x + 5z 2x 2 – 6x, 5xy + 3x 3x 3 + 4x 2 + 9xy 2 Trinomial x + y + 4z3x 2 – 4x + 67x 3 + 5X xy 2 + x 2 + 4

 Start with 3x x 5 – 7x 3  Put them in order, from largest degree to smallest Leading Coefficient Degree 2x 5 – 7x 3 + 3x 2 - 3

1. Discuss geometric representations of terms  “5” means 5 units, a rectangle 1 unit by 5 units  “2x” means a rectangle “x” units by 2 units  “x 2 ” means a square “x” units by “x” units  “x 3 ” means a cube with “x” on a side 2. Eliminate any parenthesis by distribution property 3. Combine like terms 4. Make and play with algebra tiles:  Add (x 2 + 2x – 3) to (x 2 + 4x – 2)  Subtract (2x + 3) from (x 2 + 3x + 5)  Subtract (2x + 3) from (x 2 + x + 1)

Page 61, # 1 – 15 odds (8 problems)

 Get in groups of 2 to 3 and solve problem 16 on page 61: For 1995 through 2005, the revenue R (in dollars) and the cost C (in dollars) or producing a product can be modeled by Where t is the number of years since Write an equation for the profit earned from 1995 through (hint: Profit = Revenue – Cost), and calculate the profit in 1997.

 Go over problem 15 on page 61.  Domain – independent variable – you choose – the x-axis.  Range – Dependent variable – you calculate – the value depends on what you used for the independent variable – the y-axis.  What is the domain and range of the problem?

 Multiply using distributive property. NOTE: This always works!!!!!  Multiply (2x + 3) by (x + 2) 2x(x + 2) + 3(x + 2) 2x 2 + 4x + 3x + 6 Answer: 2x 2 + 7x + 6  Show how to multiply with algebra tiles  Show how to do it on the Cartesian Coordinate Plane

 Multiply using distributive property. NOTE: This always works!!!!!  Multiply (2x - 3) by (-x – 2) 2x(-x – 2) – 3(-x – 2) -2x 2 – 4x + 3x + 6 Answer: -2x 2 – x + 6  Show how to multiply with algebra tiles  Show how to do it on the Cartesian Coordinate Plane  Multiplication of more complicated expressions are hard to show on algebra tiles and Cartesian Coordinate plane.

 Example: Multiply x 2 - 2x + 5 by x + 3 using the distributive property x(x 2 - 2x + 5) + 3(x 2 - 2x + 5 ) x 3 - 2x 2 + 5x + 3x x 3 + x 2 – x + 15  Show how to multiply on a table: x2x2 -2x+5 x +3

 Example: Multiply x 2 - 2x + 5 by x + 3 using the distributive property x(x 2 - 2x + 5) + 3(x 2 - 2x + 5 ) x 3 - 2x 2 + 5x + 3x x 3 + x 2 – x + 15  Show how to multiply on a table: x2x2 -2x+5 xx3x3 -2x 2 +5x +3+3x 2 -6x+15

 Example: Multiply x 2 - 2x + 5 by x + 3 using the distributive property x(x 2 - 2x + 5) + 3(x 2 - 2x + 5 ) x 3 - 2x 2 + 5x + 3x x 3 + x 2 – x + 15  Show how to multiply on a table: x2x2 -2x+5 xx3x3 -2x 2 +5x +3+3x 2 -6x+15 x3x3 +x 2 -x+15

Page 66, # 3 – 18 by 3’s and 19 – 22 all (10 problems)

 You learned in Algebra 1A some patterns, some special products of polynomials. You need to remember/memorize them:  (a + b) 2 = a 2 + 2ab + b 2  (a – b) 2 = a 2 – 2ab + b 2  (x + a)(x + b) = x 2 + (a + b)x + ab  (a + b)(a – b) = a 2 – b 2  NOTE: You can multiply these together by the distribution property, but you will be required to “go the other way” when we get to factoring so please learn them now.

Page 70, # 3 – 18 by 3’s (6 problems)

 So, what if we want to take a binomial and multiply it by itself again and again?  Multiply (a + b) 2  Multiply (a + b) 3  Multiply (a + b) 4  Multiply (a + b) 5

 Shortcut:  Pascal’s Triangle  Multiply:  (x + 2) 5  (3 - x) 4  (2x – 4) 3  (3x – 2y) 3

Page 75, # 12 – 17 all (6 problems)