UNIT 3 (Chapter 5 in book) – Polynomials Lesson1: The Language of Algebra

Variable: A ________ that takes the place of an unknown ___________

Monomial (or Term):

• a _______, _______, or a ________ of a number and one or more variables.

• Terms that are just numbers (without a variable) are called ____________________.

Coefficient: The numerical portion of a term; the number multiplied by the variable portion.

A polynomial is an algebraic ____________ formed by adding or subtracting _____________.

Binomial: A polynomial made up of ____ terms.

Trinomial: A polynomial made up of ____ terms.

Degree of a term: the sum of all the _________ of all its variables

Degree of a polynomial: the greatest degree of any ______ in the polynomial

Example 1: Complete this table.

# of Terms Type of Polynomial

Degree of Polynomial

a) 5y2

b) 4x + 3w2

c) 63x2y

d) x3 + 3x - 7

e) x + 5

f) 5x3 – 2x2 + 11x -9

g) 9

Modelling Polynomials

Polynomials can be modeled using _____________ _____ and ____________.

A 1-tile has dimensions _________

An x tile has dimensions _________

An x2-tile has dimensions

_________

Example 2: What expression is shown by the algebra model?

Example 3: Use a model to represent the polynomial: – 2x2 + x - 4

Class work/HW: p. 178 #1-17; Challenge #27-31

Unit 3 Lesson 2 Equivalent Expressions (5.2 in textbook) Identify the ‘coefficient’, the ‘variable’ and the ‘exponent of the variable’:

Example 1: For each of the following, identify the coefficient, the variable(s), and the exponent of each variable.

Term Coefficient Variable(s) Exponent(s) of the variable(s)

2x w3

-3ab2 -g 15

Like terms: terms that differ only by their numerical coefficients. (the variable part is the same). When you use algebra tiles, you can recognize like terms because the individual tiles have the same size and shape.

Example 2: Identify the groups of like terms in the group.

a) 4"2"% & 2" & 4 & "% & 1

b) ()2(%5(3()&3(%5)

Combining Like Terms

A _____________________ may be formed by pairing one tile with its opposite. You can _______ or _______ zero pairs without changing the polynomial.

1. Model each term. 2. Combine like terms and remove zero pairs. 1. Write the polynomial from the tiles that remain

We can also add polynomials by grouping ‘like’ terms together. The coefficients can then be added to simplify the expression.

Example 3: Combine like terms.

a) 2x2 + 2x + x2 - 3x

b) 3x2 – x + 1 + 2x – 4x2 - 5

c) x2 – 3xy + 1 + 2 – 4y2 – 5xy

Using algebraic expressions to represent problems

Example 4: Tickets for a school concert cost $10 for adults and $5 for students. Write an expression to show the total cost of tickets for a family. What do your variables represent? How much will the tickets cost for a family with 2 adults and 3 children?

Example 5: A cell phone plan charges $20 per month, plus $0.50 per text message. Write an expression to represent the total cost for one month. What do your variables represent? What will be the monthly cost if you send 35 text messages?

Class work/HW: p. 181 #26; p. 187 #2, 3, 7, 8, 10, 16, 18, 20, 21; Challenge p.189 #23-25

UNIT 3 LESSON 3 - Adding and Subtracting Polynomials Adding Polynomials – is the same as combining like terms

To add polynomials you can

1. Use algebra tiles a. Model each polynomial b. Combine like terms (same shape tiles) and remove zero pairs c. Write the polynomial from the tiles that remain

2. Combine like terms by adding the coefficients of the like terms

Example 1: add the polynomials

a) (2x2 + 2x + 3) + (x2 -3x -2)

b) (3x2 – 4x + 8) + (2x – 7x2 -5)

Subtracting Polynomials

To subtract a polynomial, just like with integers we can add the ‘opposite’ of each term in the polynomial and change it to an addition statement.

Polynomial Opposite -6x2

-5m + 3n 2y2 – 6y + 11 7a + 9b -4

Example 2: Subtract the polynomials

a) (3x2 + 2x + 3) – (x2 -3x -2) Hint: first find the opposite of (x2 – 3x -2)

b) (3n3 + 13n2 + 5n) – (7n3 + 3n2 - n)

c) (x2 + 3) + (3x2 – 2x + 6)

Class work/HW: p.196 #5, 6, 8, 11, 12, 13, 15, 16 Challenge #23, 24, 29

Unit 3 Lesson 4: Multiplying Polynomials (7.1 in book)

To determine the product of 2 monomials, we can use two methods:

a) algebra tiles (area model) b) algebra

a) Using algebra tiles:

Ex. (3x)(2x)

Make a grid, with one monomial on one dimension and the other monomial on the other dimension (length and width of a rectangle)

Use tiles of the appropriate dimensions to fill in the grid. The product is the sum of all the tiles within the grid. In this case, the product is 6x2.

b) Using algebra:

- Multiply the coefficients - Multiply the variables using exponent laws.

Ex. (2x)(3x) = (2)(3) (x)(x) = 6 x2

Example 1: Multiply the monomials

Using tiles: (3x)(2x)

Using tiles: (-x)(4x)

Using algebra: (3a)(-2b)

Using algebra: (-5x)(3.2)

(3x)(3xy)

(3.5x2)(5x3)

Example 2: Calculate the area

Class work/HW: p.260 #3, 5, 7, 10, 17 Challenge # 21, 22, 24, 26, 27

Unit 3 Lesson 5: Dividing monomials

Dividing monomials:

Using an area model, place the denominator onto one side of your grid, and then form a rectangle inside the grid with the tiles that make up the numerator:

Ex. (-8x2)/(2x)

The tiles that make up the missing dimension is the quotient.

-8x2/2x = -4x

Dividing algebraically. To divide, use exponent laws.

10x2 = 10 . (x)(x) = 5x 2x 2 (x)

TRY:

Using Tiles: (6x)/(3)

Using tiles: (6x2)/(3x)

Using Algebra: (-12x2)/4x

Using algebra: (25a)/5

(16x3)/(4x2)

(1.5x2)/(0.25x)

Class work/HW: p.261 #13 (skip b), 15, 16, 18, 19

Unit 3 Lesson 6: Multiplying Polynomials by Monomials (7.2 in textbook)

We can multiply polynomials by monomials using:

a) algebra tiles b) area models c) algebra

a) Algebra Tiles – Put the polynomial

on one dimension of a grid, and the monomial on the other using x-tiles and 1-tiles. Then, fill in the area using x2-tiles, x-tiles and 1-tiles to find the product.

b) Using an area model – instead of using tiles, write the symbols as the dimensions for the area of a region. Add the areas of each region to find the total area (the product).

c) Distributive Method – multiply each term in the polynomial by the monomial.

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

TRY:

Using Tiles: (2x)(x-1)

Using Tiles (-x)(3x+5)

Using Area Model: (-3x)(2x+1)

Using A.M. : (2x)(-3x-8)

Using D.M.: (-5)(x+5)

Using D.M: (+12x)(-6x+5)

(+5x2)(x2-2x)

(3.2x)(2x + 1.3)

Class work/HW: p.268 #2-13, 18, Challenge: #19-21

Unit 3 Lesson 7: Dividing Polynomials by Monomials (7.3 in textbook)

We can divide polynomials by:

a) Using Algebra Tiles b) Algebraically

a) Using Algebra tiles - Place the denominator along one dimension of a grid, and then fill in the grid with the tiles representing the numerator so that it forms a rectangle with the same width as the denominator. The other dimension formed represents the quotient.

b) Algebraically – Each term in the polynomial can be divided separately by the denominator, and then each term can be simplified.

-2x2 + 6x = -2x2 + 6x = -x + 3 2x 2x 2x

TRY:

Tiles: (-6x2 + 3x) / (3x)

Tiles: (2y2 + 4y) / (-2y)

(16x2 – 12x) / (4x)

(-15x2 – 5x) / (-5)

(4x3 – 3x2) / (x2)

(1.6x2 – 3.2x) / (0.8x)

Problem: The area of Rectangle B is three times the area of Rectangle A. Provide each of your answers in simplified form.

a) Write an expression to represent the area of Rectangle A. b) Determine an expression for the area of rectangle B. c) If the width of Rectangle B can be represented by 12x, what is an expression

for the length of Rectangle B?

Class work/HW: p.275 #2, 4, 6, 8, 9, 11, 13, Challenge #14-18

Unit 3 Lesson 8: Multiplying binomials by binomials How many different ways can we calculate the area of the rectangle below?

This is the area represented by the dimensions (x+3)(x+1). What product does the area represent?

How do we represent (X+5)(x-2)? What is the product?

How do we use the distributive property when multiplying two binomials?

Foil Method: F_______, O_______, I_______, L_______

Find the product for (x+5) (x+3)

F O I L

TRY:

a) (x+3) (x-2)

b) (x-1) (x-3)

c) (2x+3) (x+2)

d) (2x-3)(3x-1)

e) (x-6)2

f) (2x+7)2

g) Challenge (x+2) (x2 + 3x + 2)

h) Challenge (x+1) (5x+3) + (3)(2x+4)(6x-2)

Class work/HW: Multiplying polynomials worksheet