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.

What is a Monomial?

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:

x x-1 x1/2 2x 1 log x ln xy x2

sin x cos x tan x

POLYNOMIALS (2.1)Monomial, Binomial, Trinomial - # of termsDegree – 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 possibilitiesDegree# Terms

Zero First Second Third

Monomial

Binomial

Trinomial

POLYNOMIALS (2.1)Monomial, Binomial, Trinomial - # of termsDegree – 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 possibilitiesDegree# Terms

Zero First Second Third

Monomial

25 x, 3y, 4z 3x2, 4xy 6x3, xy2

Binomial 2 + 5 2x + 4, 3x + 5z

2x2 – 6x,5xy + 3x

3x3 + 4x2 + 9xy2

Trinomial 3 + 6 - 7

4x + y + 4z

3x2 – 4x + 6

7x3 + 5X2 + 4

2xy2 + x2 + 4

Start with 3x2 - 3 + 2x5 – 7x3

Put them in order, from largest degree to smallest

Leading Coefficient Degree

2x5 – 7x3 + 3x2 - 3

Add, Subtract Polynomials MM1A2c, Section 2.1

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 “x2” means a square “x” units by “x” units “x3” means a cube with “x” on a side

2. Eliminate any parenthesis by distribution property

3. Combine like terms4. Make and play with algebra tiles:

Add (x2 + 2x – 3) to (x2 + 4x – 2) Subtract (2x + 3) from (x2 + 3x + 5) Subtract (2x + 3) from (x2 + x + 1)

Addition and Subtraction Assessment

Page 61, # 1 – 15 odds (8 problems)

Warm-Up 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 1995. Write an equation for the profit earned from 1995 through 2005. (hint: Profit = Revenue – Cost), and calculate the profit in 1997.

4004

21

4

1 2 ttR 2004

13

12

1 2 ttC

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 Polynomials MM1A2c, Section 2.2

Multiply using distributive property. NOTE: This always works!!!!!

Multiply (2x + 3) by (x + 2)2x(x + 2) + 3(x + 2)2x2 + 4x + 3x + 6

Answer: 2x2 + 7x + 6 Show how to multiply with algebra tiles Show how to do it on the Cartesian

Coordinate Plane

Multiply Polynomials (2.2) Multiply using distributive property.

NOTE: This always works!!!!! Multiply (2x - 3) by (-x – 2)

2x(-x – 2) – 3(-x – 2)-2x2 – 4x + 3x + 6

Answer: -2x2 – 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.

Multiply Polynomials (2.2) Example: Multiply x2 - 2x + 5 by x + 3

using the distributive propertyx(x2 - 2x + 5) + 3(x2 - 2x + 5 )

x3 - 2x2 + 5x + 3x2 - 6 + 15

x3 + x2 – x + 15 Show how to multiply on a table:

x2 -2x +5

x+3

Multiply Polynomials (2.2) Example: Multiply x2 - 2x + 5 by x + 3

using the distributive propertyx(x2 - 2x + 5) + 3(x2 - 2x + 5 )

x3 - 2x2 + 5x + 3x2 - 6 + 15

x3 + x2 – x + 15 Show how to multiply on a table:

x2 -2x +5

x x3 -2x2 +5x+3 +3x2 -6x +15

Multiply Polynomials (2.2) Example: Multiply x2 - 2x + 5 by x + 3

using the distributive propertyx(x2 - 2x + 5) + 3(x2 - 2x + 5 )

x3 - 2x2 + 5x + 3x2 - 6 + 15

x3 + x2 – x + 15 Show how to multiply on a table:

x2 -2x +5

x x3 -2x2 +5x+3 +3x2 -6x +15x3 +x2 -x +15

Multiplication Assessment

Page 66, # 3 – 18 by 3’s and 19 – 22 all

(10 problems)

Special Products of PolynomialsMM1A2e, Section 2.3

You learned in Algebra 1A some patterns, some special products of polynomials. You need to remember/memorize them:

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

(x + a)(x + b) = x2 + (a + b)x + ab (a + b)(a – b) = a2 – b2

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.

Special Products Assessment

Page 70, # 3 – 18 by 3’s (6 problems)

Binomial Theorem Not a standard, Section 2.4

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

Binomial Theorem

Shortcut: Pascal’s Triangle Multiply:

(x + 2)5

(3 - x)4

(2x – 4)3

(3x – 2y)3

Binomial Theorem Assessment

Page 75, # 12 – 17 all

(6 problems)