unit 1: chapter 3 factors & productsperfect cubes, then determine square roots and cube roots ....

Foundations of Mathematics & Pre-Calculus 10 Unit 1: Chapter 3

edited and revised by Ms Moon

Unit 1: Chapter 3 Factors & Products Big Ideas

• Arithmetic operations on polynomials are based on the arithmetic operations on integers, and have similar properties.

• Multiplying and factoring are inverse processed, and a rectangle diagram can be used to represent them.

Competency Focus: Communication and Representing

• Use mathematical vocabulary and language to contribute to mathematical discussions. • Develop mathematical understanding through concrete, pictorial, and symbolic

representations.

Learning Outcome Course Concepts Self-Assessment Pre Post

3A: I can demonstrate an understanding of factors of whole numbers by determining the:

• Prime Factors • Greatest Common Factor • Least Common Multiple • Square Root • Cube Root

I can determine prime factors, greatest common factors, and least common multiples of whole numbers.

I can identify perfect squares and perfect cubes, then determine square roots and cube roots

3B: I can demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially and symbolically.

I can model the factoring of polynomial.

I can factor a common factor from a polynomial.

I can factor a polynomial of the form 𝑥𝑥2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐

I can factor a polynomial of the form 𝑥𝑥2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐

I can factor special polynomials (perfect square trinomials and difference of squares)

3C: I can demonstrate an understanding of the multiplication of polynomial expressions, concretely, pictorially and symbolically.

I can extend the strategies for multiplying binomials to multiply polynomials.

Vocabulary

prime factorization

greatest common factor

least common multiple perfect cube cube root radicand

radical index factoring decomposition

perfect square trinomials

difference of squares



Name: ____________________________

Due Date: _________________________

Unit 1: Chapter 3 Factors & Products Assignment List

Section Page #s / Question #s Reading Page #s

Completed / Checked

3.1 pp.140–141 #3ace–11ace. 12, 14, 16aceg pp.142–148

3.2 pp.146–147 #4ace–6ace, 7, 8, 10, 15, 16 pp.150–156

3.3 pp.155–156 #4–6, 7ace–10ace, 14ac, 16ace pp.157–167

Review p.149 #1–10; p.180 #1–2; pp.198–199 #1–14

Quiz 3.1-3.3

3.5 pp.166–167 #3, 4, 11ace, 14ace, 17, 19ace–21ace

pp.168–181

3.6 pp.176–178 #5ac, 9ace, 10ace, 13ace, 18ac, 19ace, 20a

pp.182–187

3.7 pp.185–187 #4ac, 5ace, 13c, 15ce, 17, 18ac, 21a, 22d

pp.188–196

3.8 pp.194–195 #5, 6, 10ace, 11ace–13ace, 18, 21ace

Review pp.180–181 #5–9; pp.199–200 #18–35

Quiz 3.5-3.8

Review p.201 #1–9; pp.252-253 #7–18

Chapter Test

Assignment: Read pp.134 – 141; read everything including The World of Math



Section 3.1 Factors and Multiples of Whole Numbers (pages 134 – 141) Number Systems Natural Numbers:

Whole Numbers:

Integers:

Rational Numbers:

Irrational Numbers:

Real Numbers:

Divisibility Properties Divisibility by 2: A whole number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8

Divisibility by 3: A whole number is divisible by 3 if the sum of its digits is divisible by 3

Divisibility by 4: A whole number is divisible by 4 if the last two digits are divisible by 4

Divisibility by 5: A whole number is divisible by 5 if the last digit is 0 or 5

Divisibility by 6: A whole number is divisible by 6 if it is an even # that is divisible by 3

Divisibility by 9: A whole number is divisible by 9 if the sum of its digits is divisible by 9

Divisibility by 10: A whole number is divisible by 10 if it ends in a 0



Prime Number:

Factor:

Multiple:

Composite Number:

Zero and One: The whole numbers 0 and 1 are neither prime nor composite.

Finding Prime Factors of Composite Numbers Method 1: Factor Tree

72

72 72

Method 2: Dividing by Primes until the Quotient is a Prime Numbers

Method 3: Continuous Division



Try: Write the prime factorization of 3300 Greatest Common Factor (GCF):

Finding the Greatest Common Factor Method 1:

1. Find the prime factors of each number 2. List each common factor the least number of times it appears in any one

number (select all of the prime factors common to both) 3. Find the product of those factors

27

54 27 = 54= GCF (27, 54) =

Method 2:

1. Divide by a prime factor that can divide two or more of your numbers.

2. Repeat, until no remaining numbers have any prime factors in common

) 27 54 GCF (27, 54) =

Try: Determine the greatest common factor of 126 and 144.



Least Common Multiple (LCM):

Finding the Least Common Multiple Method 1:

1. Write multiples of each number. 2. Select the smallest multiple common to both.

15: 20: LCM (15, 20) = Method 2:

1. Write each number as the product of prime factors 2. Select all of the prime factors from the first number and then select only those

prime factors from the second that are not already there 3. Find the product of those factors.

15

20 15 = 20 = LCM (15, 20) =

Method 3:

1. Divide by a prime factor that can divide two or more of your numbers. Bring down any number that does not factor

2. Repeat, until no remaining numbers have any prime factors in common

) 15 20 LCM (15, 20) =

Try: Determine the least common multiple of 28, 42, and 63.



Solving a Problem by GCF or LCM Two buses travel from a chalet to the ski hill and back several times each day. It takes bus A 20 minutes for a return trip and bus B 30 minutes. If they both start at 8:00 am, at what time will they both be back at the chalet together? Mercury, Venus and Earth revolve around the Sun every 3, 7, and 12 months respectively. If the three planets are currently lined up, how many months will pass before this happens again? There is a unique way of finding the LCM of two numbers. Take the product of both numbers, and divide that product by the GCF. Find the LCM of 18, 24 by this method. Assignment: pp.140 – 141 #3ace – 11ace, 12, 14, 16aceg Read pp.142 – 148



Section 3.2 Perfect Squares, Perfect Cubes, and Their Roots (pages 142 – 148) Perfect Squares: a whole number that can be used to represent the

area of a square with a whole number side length. Square Root: the side length of the square. The symbol √ (called

a radical sign) is used to indicate square roots

Perfect Cubes:

Cube Root:

Determining the Square Roots of a Whole Number Method 1: Factor Tree

196

Method 2: Continuous Division |196

x x2 x3

1 2 3 4 5 6 7 8 9 10 11 12 1728

13 2197

14 2744

15 3375

16 4096

17 4913

18 324 5832

19 361 6859

20 400 8000

21 441 9261

22 484 10648

23 529 12167

24 576 13824

25 15625

Side

Length

Volume;

Perfect Cube

Area;

Perfect

Square



Try: Determine the square root of 1296. Try: Determine the square root of 100441

. Determining the Cube Roots of a Whole Number Method 1: Factor Tree

216

Method 2: Continuous Division |216

Try: Determine the cube root of 2744. Try: Determine the cube root of 64

3375.



Using Roots to Solve a Problem A cube has a volume of 216 cm3. Determine the length of each side of the cube. The area of a baseball diamond is 8100 ft2.

a) How far is it from home plate to first base?

b) How far is it from 1st base to 3rd base? Try: The volume of a sphere is given by the formula V = 4

3𝜋𝜋𝑟𝑟3, with r the radius of the

sphere. Determine the radius of a sphere with volume 972 𝜋𝜋 mm3. Assignment: pp.146 – 147 #4ace – 6ace, 7, 8, 10, 15, 16 Read pp.150 – 156; copy the diagram on page 148



p.148

Properties of Whole Numbers



Section 3.3 Common Factors of a Polynomial (pages 150 – 156) Representing Multiplication using an Area Model Multiplication can be represented as an area model.

A rectangle drawn below has a dimension of _____ by ______. The

rectangle has an area of _______ squares. We know that since it

contains _______ rows of ________ squares.

Try: Draw a least 3 different models to represent the factoring of 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representing Polynomials using Algebra Tiles Term: a number, a variable, or a product of a number with one or more variables

that can be raised to a power. Polynomial: a term or a group of terms separated by additions or subtractions in

which all the variables have whole number exponents. Polynomial with one term is called

two terms is called

three terms is called

The process of representing a quantity using multiplication is called .



Using Algebra Tiles to Factor Binomials Method 1: Using Algebra Tiles

Form a rectangle with these tiles.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method 2: Use Greatest Common Factor

1. Factor each term of the binomial 2. Write each term as a product of the greatest common factor and another

monomial. 3. Use the distributive property to write the sum as a product.

Factor the following binomial: 12 − 8𝑥𝑥 Factoring Trinomials

12𝑥𝑥2 − 8𝑥𝑥 + 16 Method 1: Algebra Tiles Divides tiles into equal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Method 2: GCF Find the Greatest Common Factor



Try: Factor: 3𝑥𝑥2 − 6 12𝑥𝑥2 + 8𝑥𝑥

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Try: 5 − 10𝑎𝑎 − 5𝑎𝑎2



Factoring Polynomials in More than one Variable −12𝑥𝑥3𝑦𝑦 − 20𝑥𝑥𝑦𝑦2 − 16𝑥𝑥2𝑦𝑦2

Try: Factor: 10𝑎𝑎3𝑏𝑏2 + 8𝑎𝑎𝑏𝑏3 + 2𝑎𝑎𝑏𝑏4 25𝑥𝑥𝑦𝑦2𝑧𝑧3 − 20𝑥𝑥2𝑦𝑦4𝑧𝑧2 + 30𝑥𝑥4𝑦𝑦2𝑧𝑧5 Assignment: pp.155 – 156 #4 – 6, 7ace – 10ace, 14ac, 16ace, Reflect Read pp.157 – 167

*** QUIZ 3.1 - 3.3 tomorrow *** Reflection: How are the processes of factoring and expanding related?

Note: When we factor a polynomial that has negative terms, we usually ensure the first term inside the brackets is not negative. In order to do this we can remove a factor of ____.



Section 3.5 Polynomials of the Form x2 + bx + c (pages 159 – 167) Reviewing Trinomials of the Form x2+bx+c a)

b) c)

Multiplying Two Binomials (Expanding) A binomial product is a multiplication of two binomials. They can be written in the

general form of .

Method 1: Expand pictorially using algebra tiles/area model

1. Sketch a rectangle with dimensions corresponding to the two factors. 2. Divide the dimensions into different sections base on the type of term present. 3. Calculate the area of each section. 4. Add and combine like terms

(x +3) (x +1) Method 2: Expand algebraically using the distributive property

1. Multiply the First terms. 2. Multiply the Outside terms. 3. Multiply the Inside terms. (a + b) (c + d) = 4. Multiply the Last terms.

(x +6) (x +4)



Try: Expand

a) (x – 7) (x +2) b) (3x +1) (x – 5) c) (6a – 5b) (6a – 5b) Factoring Trinomials The two forms of the same polynomial can be represented by (x + p) (x + q) = x2 + bx + c where

b =

c =

We may use the above rule to factor algebraically.

1. Find two numbers which multiplies to the constant, c. 2. Determine which pair of products also add/subtract to yield the coefficient, b. 3. Write the binomial products with the appropriate signs.

a) x2 + 13x +12 b) x2 – 11x +10

c) y2 + 3y +4 d) q2 – 29q + 100

e) q2 + 17q +42 f) d2 + 12d – 32 Notice, not all trinomials are factorable. Pictorially speaking, trinomials that are not factorable cannot be arranged into a rectangle.



Factoring a Trinomial with a Common Factor and Binomial Factors Factor the polynomial expressions by first removing a common factor.

a) 4x2 – 32x + 48 b) 3x3 + 21x2 +30x

c) 2x2 + 6x + 4 d) -2a2 – 30a – 108

e) ax2 – 14ax + 45a f) -10a4 + 100a3 – 240a2 Assignment: pp.166 – 167 #3, 4, 11ace, 14ace, 17, 19ace – 21ace Read pp.168 – 181 Reflection: Does the order in which the binomial factors are written affect the solution?

Explain.



Section 3.6 Polynomials of the Form ax2 + bx + c (pages 168 – 181) Multiplying Two Binomials Method 1: Algebra Tiles (5e + 3) (2e + 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method 2: Area Model (2t – 9) (7 – 5t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Method 3: Distributive Property (–3k + 5) (2 – 7k)

Factoring Trinomials of the Form ax2 + bx + c Method 1: Algebra Tiles 3x2 – 13x – 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Method 2: Bring Up Method – easiest, straight forward, but long a) 8x2 – 14x + 3

1. Bring the first coefficient to the last

term and multiply them 2. Factor by looking for two numbers

that multiply to the last term but add to the coefficient in front of the x term.

3. Divide both factors by the initial coefficient in front of the x2 term.

4. Reduce and “Bring Up” the denominator.

b) 2n2 + 13n + 6

c) 15x2 + 4x – 4

Method 3: Criss – Cross Method – Fast, quick with numbers, but harder a) 72x2 + 11x – 6

1. Find two numbers that multiply to the coefficient in front of x2 and two numbers that multiply to the last term.

2. Multiply sideways or criss – cross so that the sum equals to the coefficient in front of x.

3. Place the number on the left in front of each bracket.

4. Numbers multiplied together cannot go into the same bracket.

b) 8x2 – 36x + 15

c) 6x2 – 17x + 5



Method 4: Decomposition (Grouping) Method – Textbook, standard method a) 8x2 + 14x + 3

1. Multiply the first coefficient and the last term

2. Find two numbers that multiply to the number you found but add to the coefficient in front of the x term.

3. Split the x into two parts.

4. Group the first two terms together and the last two terms together.

5. Factor out any common factors from each bracket.

6. The binomial is a GCF, factor it out.

b) 4x2 + 4x – 15

c) 3x2 – 11x – 4

Assignment: pp.176 – 178 #5ac, 9ace, 10ace, 13ace, 18ac, 19ace, 20a Read pp.182 – 187 Reflection: Will decomposition work if the a value of a trinomial is 1? Do an example to

prove this.



Section 3.7 Multiplying Polynomials (pages 182 – 187) Multiplying Polynomials The fundamental principal guiding the multiplication of polynomials is the . It states that each term of one polynomial must be multiplied by each term in all other polynomials being multiplied together. Ex. 𝑎𝑎 ( 𝑏𝑏 + 𝑐𝑐 + 𝑑𝑑 ) = ( 𝑎𝑎 + 𝑏𝑏 )( 𝑐𝑐 + 𝑑𝑑 ) = ( 𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 )( 𝑑𝑑 + 𝑒𝑒 + 𝑓𝑓 ) = Method 1: Area Model then Simplify

a) (2k2 – 3k + 1)(k2 + 4k – 5)

1. Draw an area model. 2. Label the length and width 3. Multiply 4. Simplify by combining like terms

Method 2: Distributive Property – expand the brackets then simplify

b) (2x – 4y)(4x + 3y + 5)

1. Multiply each term in the first bracket by each term in the 2nd bracket.

2. Simplify by combining like terms

c) 4m (2n3 – 5n2 + 7n – 1) d) (2x – 1)3



The following problems are slightly more challenging. They require the order of operations:

• Multiply to get rid of the brackets • Add or subtract as stated • Simplify by collecting like terms.

e) (x + 2)(x – 4) + 2 (x – 7)(x + 1)

f) 3(x + 2)(5x – 6) – 2x (3x3 – 4x + 1)

g) Determine the area of the following shaded region. 3x + 6 2x 5x + 2 4x + 14 3x + 4 5x + 10 9x + 12 Assignment: pp.185 – 187 #4ac, 5ace, 13c, 15ce, 17, 18ac, 21a, 22d Read pp.188 – 196



Section 3.8 Factoring Special Polynomials (pages 188 – 196) Perfect Square Trinomials Write the following algebra tile models as a multiplication of binomials using the template below. a) = ( ) ( )

b) = ( ) ( )

Perfect square trinomials can be drawn as a square area model or can be remembered as two generic formula: (a + b)2 = (a + b) (a + b) = (a – b)2 = (a – b) (a – b) = Difference of Squares Trinomials Write the following algebra tile models as a multiplication of binomials using the templates below. c) = ( ) ( )

d) = ( ) ( )

Difference of squares trinomials can be drawn as an area model which is a subtraction of two squares or can be remembered as

a2 – b2 = ( ) ( ). Expanding binomial products Expand the following binomial products. Identify the type of special polynomials.

a) (2x + 7)2 b) (a2 – 5)2

c) (4 – k)(4 + k) d) (3m2 – 4n)(3m2 + 4n)



Factoring Special Polynomials

a) 121x2 – 22x + 1 b) 4x2 – 9

c) 4x2 + 4xy +y2 d) 16a2 – 25

e) 25v2 – 70vw +49w2 f) 49m4 – 36n2

g) 9x4 – 13x2 + 4 * h) 4x4 – 81x2 * Assignment: pp.194 – 195 #5, 6, 10ace, 11ace – 13ace, 18, 21 ace

*** QUIZ 3.5 - 3.8 tomorrow *** *****Chapter 3 Test on ___________________*****

Reflection: Does a sum of squares factor? Explain.



Factoring Flow Chart

TWO Probably a difference of squares: * You need subtraction (“difference”) and squares 𝑎𝑎2 − 𝑏𝑏2 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 − 𝑏𝑏)

Conjugate = opposite sign in the middle of two terms Example: 4𝑥𝑥2 − 9 = (2𝑥𝑥)2 − (3)2 = (2𝑥𝑥 + 3)(2𝑥𝑥 − 3)

THREE Factoring trinomials:

𝑎𝑎𝑥𝑥2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐 Type 1: a = 1 Example:

𝑥𝑥2 − 3𝑥𝑥 + 2 Ask: What ADDS to “b” (here – 3 ) & MULTIPLIES to “c”

(here + 2 ) Answer: – 1, – 2 Type 2: a≠ 1 Example:

2𝑥𝑥2 − 𝑥𝑥 − 1 Ask: What ADDS to “b” (here – 1 ) & MULTIPLIES to “ac” (here 2(– 1) = – 2 ) Answer: – 2, 1 Use these to split the middle term into two separate terms:

2𝑥𝑥2 − 𝒙𝒙 − 1 2𝑥𝑥2 − 𝟐𝟐𝒙𝒙 + 𝟏𝟏𝒙𝒙 − 1

Factor using grouping: See next column 😊😊

FOUR Probably grouping: Example:

2𝑥𝑥2 − 2𝑥𝑥 + 1𝑥𝑥 − 1 Grouping the first two terms together, and the last two terms together:

[2𝑥𝑥2 − 2𝑥𝑥] + [1𝑥𝑥 − 1] Factor common factors out of each group:

2𝑥𝑥(𝒙𝒙 − 𝟏𝟏) + 1(𝒙𝒙 − 𝟏𝟏) You should have two matching brackets. Factor them out:

(𝒙𝒙 − 𝟏𝟏)(2𝑥𝑥 + 1)

STEP 3: Ask: FF? Look inside each factor (bracket) & see if you can FACTOR FURTHER.

STEP 1: Take out COMMON FACTORS (GCF)

STEP 2: Ask: How many terms are there?

unit 1: chapter 3 factors & productsperfect cubes, then determine square roots and cube roots ....

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