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Math 8 - Unit 14

Polynomials and Factoring

Extra Help: Tuesday, 5/12, 2:40pm * Tuesday, 5/21, 7:30am * Friday, 5/31, 7:30am

Name: _________________________

Date Lesson Topic Homework

F 5/10 1

Review: Adding, Subtracting and

Multiplying Polynomials

Lesson 1 – Page 7

M 5/13 2

Multiplying a Binomial by a Binomial

Day 1

Lesson 2 - Page 11

T 5/14 3

Multiplying a Binomial by a Binomial

Day 2

Lesson 3 – Page 14

W 5/15 4

Multiplying a Binomial by a Polynomial

Lesson 4 – Page 17 and 18

Th 5/16 *FIELD TRIP *

F 5/17 UNIT QUIZ

M 5/20 5 Greatest Common Factor Lesson 5 – Page 22

T 5/21 6 Factoring Out Like Terms Lesson 6 – Page 25

W 5/22 *SAGAMORE DAY *

5/23-

5/28

MEMORIAL DAY WEEKEND! Enjoy the long holiday weekend 😊

W 5/29 7

Factoring Trinomials

-Two Sums and Two Differences

Lesson 7 – Page 27

Th 5/30 8

Factoring Trinomials

-One Sum and One Difference

Lesson 8 – Page 29

F 5/31 9

Factoring Trinomials

-Mixed Practice

No Homework

M 6/3 Unit Review Study!

T 6/4 UNIT TEST

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Lesson 1 – Adding, Subtracting and Multiplying Polynomials

Aim: I can add, subtract and multiply polynomials.

Warm Up: Simplify the following expressions (combine like terms)

1) 6x + 3x 2) 5x + x 3) x2 + 7 4) -9x + (-4x)

5) -2x – 11x 6) 3x2 + x2 7) -3x – 2x 8) 10y – (- 3y)

9) 9y – 3 + 6y – 8 10) 9x + 4 11) -7x + 7x 12) 2a + 5b – 4a + 3b

Simplify:

13) Distribute: - (3x2 + 2x – 6) 14) Subtract 8y2 from 2y2

Vocabulary:

Expressions vs. Equations ________________________________________________________________

Identify the parts of an expression

3x2

Numerical coefficient ______ Base______ Exponent______ Variable ______

Polynomial ___________________________________________________________________________

Standard form (ax2 + bx + c) _____________________________________________________________

Degree of a polynomial __________________________________________________________________

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Guided Practice:

A. Put the following polynomials in standard form and state it’s degree.

1) -4x2 + 5x + 3x3 – 9 2) 4x – 9x2 + 3 Degree = _____ Degree = ______

B. Add the following polynomials.

1) (4x2 – 3x + 2) + (5x – x2 – 1) Rule : Remove Parentheses from BOTH Expressions and Combine Like Terms. Arrange in Standard Form.

2) (4x – 9x2 + 3) + (x2 + 5x – 4)

C. Subtract the following polynomials.

1) (4x2 – 3x + 2) – (5x + x2 – 1) Rule : Remove Parentheses from the FIRST Expression and DISTRIBUTE the subtraction sign into the SECOND Expression. Combine Like Terms. Arrange in Standard form.

2) (4x – 9x2 + 3) – (x2 + 5x – 4)

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D. Multiply the following polynomials

1) x2 • x3 2) 3(6𝑐 + 3𝑑)

3) 9x(4x + 2) 4) x (9x2 + 4x + 3)

Rule : Multiply the coefficients together. Multiply the like variables (keep the base and add the exponents) Arrange in standard form.

Independent Practice: Simplify the expressions

1) (2𝑥2 – 4) + (x2 +3𝑥 − 3) 2) (2b2 – 11b + 2) + (3b2 + 11b – 2)

3) (3a2 – 6a + 8) – (4a2 – 7a + 12) 4) 𝑥 – (3𝑥 – 4)

5) Subtract 𝟐𝒙𝟐 + 𝟓𝒙 – 𝟑 from 𝒙𝟐 – 𝟕𝒙 + 𝟑

6) Find the perimeter of a triangle whose sides measure: 3𝑥 + 5, 𝑎𝑛𝑑 2𝑥 + 9, and 5𝑥 + 𝑦.

7) 12x (12x + 11) 8) -9x (-3x2 + 9x + 11)

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Multiply polynomials.

1) )32(6 2 +− x 2) )7( 2 xxx − 3) )23( 354 xxx −−

4) 3h(5h3 – 6h) 5) 2x(3x3 - x2 – 5) 6) -2n (3n2 – 3n – 7)

7) w2 (5w3 + 7w – 3)

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Lesson 1 Homework

Simplify and express in standard form.

1) (4a2 – 8a – 5) + (a2 – 2a – 12) 2) -6y2 + 3y – 1 – 9y2 – 5y + 2

3) (7x2 - 2x – 6) + (x2 + 2x + 9) 4) -x2 + 1 – 7x2 – 11

5) (-5x)(6x) 6) x(x – 4) 7) (x - 4)3

8) )32(6 2 +− x 9) )7( 2 xxx − 10) )23( 354 xxx −−

11) 3h(5h3 – 6h) 12) 2x(3x3 - x2 – 5) 13) -2n (3n2 – 3n – 7)

14) w2 (5w3 + 7w – 3) 15) 3(b3 +8b) – 2(b3 + 12)

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Lesson 2 – Multiplying Binomials

Aim: How do we multiply a binomial by a binomial?

Warm up: Simplify

a) (2x2 – 4) + (x2 + 3x – 3) b) (9b2 + 4b) – (b – 8)

c) Analyze Timothy’s answer below:

Describe and correct the error in finding the difference of the polynomials.

d) Samantha was asked to find the sum of −5𝑥2 + 1 and 2𝑥 − 8. Below is her answer:

Determine if Samantha’s answer is correct. Be sure to justify your reasoning.

Timothy’s Work

(x2 - 5x) – (-3x2 + 2x) = (x2 - 5x) + (3x2 + 2x)

= (x2 + 3x2) + (-5x + 2x)

= 4x2 – 3x

−5𝑥2 + 1

+ 2𝑥 − 8

−𝟑𝒙 − 𝟕

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Guide Practice:

Double Distribute using a box (Diagram) (x + 2)(x + 3)

Final Answer: x2 + 5x + 6

Examples:

1) (x + 1)(x + 4) 2) (x + 2)(x + 5)

3) (𝑥 + 3)(𝑥 − 2)

4) (𝑥 + 5)2

x2

3x

2x

6

x

2

2

x 3

x 1

x

4

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5) (3x − 5)(4x − 2)

Independent Practice:

1. (𝑥 + 2) (𝑥 − 1)

2. (𝑧 + 4)(𝑧 + 3)

3. (5𝑦 − 2) (3𝑦 + 1)

4. Find the area of a square who’s side measures (𝑥 − 3).

5. Determine whether the following statement is true: (𝑎 + 𝑏)2 = 𝑎2 + 𝑏2. Justify your reasoning.

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Lesson 2 Homework

1) (x + 3)(x + 8) 2) (x - 4)(x - 8) 3) (x + 2)(x - 8)

4) (x + 5)(x – 4) 5) (x + 2)(x + 7) 6) (x – 3)2

7) x(x) 8) 7x2(3x2) 9) (74)5 10) (8x7)2

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Lesson 3 – Multiplying Binomials

Aim: I can multiply a binomial by a binomial

Find the missing number:

1) (x + 1)(x + 4) = ____ + 5x + 4 2) (x + 4)(x + 5) = x2 + ____ + 20

3) (x + 5)(x - 2) = x2 + 3x - ____ 4) (x + 2)(x - 8) = x2 - ____ - 16

5) (x - 3)(x - 4) = x2 - ____ + 12 6) (x - 5)(x - 9) = x2 - 14x + ____

Guided Practice:

1) (3x + 1)(3x+ 4) 2) (2x + 7)(3x - 3)

3) (2x + 7)(2x - 7) 4) (4x – 3)(2x – 4)

5) (4x – 1)(2x + 3) 6) (2x – 6)(3x – 9)

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Independent Practice:

1) (3x + 8)(2x – 2) 2) (5x + 2)(2x – 4)

3) (3x – 4)(x + 5) 4) (5x + 7)(x – 3)

5) (2x + 1)2 6) (6x – 5)(6x + 5)

7) (7x + 8)(2x – 3) 8) (4x + 9)(x + 4)

9) (8x – 4)(3x + 5) 10) (x - 2)(6x – 4)

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Lesson 3 Homework

1) (3x + 6)(3x – 6) 2) (5x – 3)(7x + 3)

3) (x + 2)(x + 8) 4) (x - 6)(x - 2)

5) (x + 4)2 10) (x + 3)(x - 3)

11) (x + 5)(x - 2) 12) (x + 1)(x - 7)

13) (5x - 3)(2x + 6) *14) 3(x + 2)(x - 7)

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Lesson 4

Multiplying a Binomial by a Polynomial

There are 2 Methods to Multiply a Binomial times a Polynomial

1) Double Distribute lining up like terms

2) Double Distribute using a box (Diagram)

Method 1: Double Distribute lining up like terms

Step 1: Multiply first term by each (x + 2)(x2 + 5x - 3)

term in the parentheses

x3 + 5x2 - 3x

+ 2x2 + 10x - 6

Step 2: Multiply the second term by (x + 2)(x2 + 5x - 3) x3 + 7x2 + 7x - 6

each term in the parentheses

Step 3: Combine Like Terms

Method 2: Double Distribute using a box (Diagram)

Using the double distributive property:

(x + 2)(x2 + 5x - 3)

x3 + 7x2 + 7x - 6

Rules:

Step 1: Distribute (multiply) the first term to each term in the second parentheses.

Step 2: Distribute (multiply) the second term to each term in the second parentheses.

Step 3: Be sure to line up LIKE terms under each other - Combine like terms.

Guided Practice:

1) (x + 4)(x2 − 3x + 5) 2) (2x + 3)(x2 − 4x – 6)

x x3 5x2 -3x

2 2x2 10x -6

x2 5x -3

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3) (x2 − 2x + 5)(x − 7) 4) (w + 1)(w2 − w + 1)

5) (x + 2)(x – 5) 6) (2y + 1)(3y2 − 4y + 2)

Independent Practice:

1) 2x4(5x3 – 3x2 + x + 15) 2) (3x − 8)(4x2 + 2x + 3)

Draw a picture to represent the expression

3) (x + 8)(3x2 + 5x - 6) 4) (3x2 + x − 1)(x − 2x + 1)

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Lesson 4 Homework

Simplify: Solve by double distributing:

1) (2x – 3)(3x2 – 5x + 4) 2) (x + 2)(x – 6)

Solve by drawing a diagram:

3) (𝑥 − 1)(𝑥2 − 𝑥 + 1)

Solve any method:

4) (3𝑥2 + 4𝑥 + 2)(2𝑥 + 3) 5) (𝑥 − 5)(𝑥2 + 𝑥 + 1)

6) (2𝑥2 + 10𝑥 + 1)(𝑥 + 1) 7) (4𝑥 + 3)(2𝑥 + 5)

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8) Application Problem: The figure below is a square. Find an expression for the area of the shaded region.

Write your answer in standard form.

9) What is the final answer using this diagram?

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Mixed Review Extra Help:

1) (8x + 2)(3x + 1) 2) (3x + 1)(x2 + 2x + 1)

3) (x + 3)2 4) (a – b)(a2 + 2ab + b2)

Draw a picture to represent the expression

5) (x + 3)(x3 – 2x2 – x + 3)

6) (x2 + x – 5)( 2x2 – x + 5)

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Lesson 5 – Greatest Common Factor

Aim: What is factoring and how do we factor polynomials? Warm Up: 2x +3 3x + 5 1. Express the perimeter of the rectangle as a binomial in simplest terms of x. 2. Express the area in of the rectangle as a binomial in simplest terms of x.

A. Factor – ________________________________________________________

____________________________________________________

B. Finding the Greatest Common Factor

1. 30,12 2. 16, 40, 28 3. 18, 15 4. 12, 20

25x4, 30xy 16xy2, 64y3z2 Numbers: Numbers:

Variables: Variables:

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Find Greatest Common Factor 5. x3, xy3 6. x3, x5 7. 3y, -8xy 8. 22x2, -11x 9. 35ab, 50b 10. 40m, 60n 11. 8x, 28x2y 12. 3x, 12 C. Factor Using GCF

1. 6c3 – 12c2 + 3c 2. 3a + 6a2 + 15a3 3. 12a2 + 20ab 4. 3x – 3y

5. xc + xd 6. 3m – 6n 7. 18c + 27d 8. 7y – 7 9. 3x2 – 6x - 30

10. 3x5 – 15y

11. 5x3 - 7x

12. 4x2 + 9x

13. 33x2 - 30x

14. 6x3 + 4x2 + 2x

15. 3x6 + 15x4

16. 3x6 - 9x5 - 15x4

17. 3x3 + 10x2 - x

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Lesson 5 Homework

Factor each polynomial.

1. 36x3 + 28x

2. 3x2 - 3x

3. -3x3 - 33x

4. -15x2 + 18x

5. 4x3 - 28x

6. 16x3 + 10x2 - 18x

7. 19x3 - 19x

8. 6x3 + 8x

9. 36x3 - 24x2 + 8x

10. 14x2 + 16x

11. 16x4 - 32x3 - 80x2

12. 14x5 - 24x4

13. x3 + 3x

14. 4x2 + 3x

Multiply

15. 3x(4x – 5) 16. 6x(x – 8) 17. 2x(x – 5)

18. What is the supplement of 60?

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Lesson 6 – Factoring Trinomials – Two Sums

Aim: How do we factor trinomials? Warm Up:

1. Simplify 3x0 2. Simplify (3x)0

3. What are factors of 18 whose sum is 11? 4. What are factors of 24 whose sum is 10?

5. What are factors of 12 whose sum is -7? 7. What are factors of 20 whose sum is-12?

Factoring Trinomials x2 + 5x + 6

Find the factors of x2 (x __)(x __)

Find the factors of 6 whose sum is 5. ( x + 2) (x + 3)

Final factored form ( x + 2 )( x + 3 )

Factor each trinomial into a binomial pair:

1. x2 + 7x + 10 2. x2 + 8x + 15 3. x2 + 8x + 7 4. x2 + 5x + 6

5. x2 + 8x + 12 6. x2 + 9x + 14 7. x2 + 5x + 4 8. x2 + 3x + 2

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9. x2 + 6x + 9 10. x2 + 9x + 20 11. x2 + 4x + 4 12. x2 + 10x + 25

13. x2 + 11x + 24 14. x2 + 12x + 35 15. x2 + 10x + 24 16. x2 + 9x + 20

17. x2 + 8x + 16 18. x2 + 4x + 3 19. x2 + 14x + 49 20. x2 + 11x + 30

21. If one factor of x2 + 9x + 18 is (x + 6), what is the other factor?

22. If one factor of x2 + 9x + 14 is (x + 2), what is the other factor?

23. If one factor of x2 + 13x + 36 is (x + 4), what is the other factor?

24. If one factor of x2 + 15x + 56 is (x + 8), what is the other factor?

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Lesson 6 – Homework

Factor each polynomial.

1. x2 + 5x + 6 2. x2 + 7x + 12 3. x2 + 8x + 12

4. x2 + 21x + 20 5. x2 + 10x + 24 6. x2 + 13x + 40

7. x2 + 11x + 24 8. x2 + 13x + 42 9. x2 + 6x + 8

10. x2 + 16x + 28 11. 5x + 5 12. 8x + 8y

13. x2 + 10x + 16 14. x2 + 9x + 18 15. 4x2 + 5x

16. If one factor of x2 + 7x + 10 is (x + 5), what is the other factor?

17. If one factor of x2 + 7x + 10 is (x + 2), what is the other factor?

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Lesson 7

Factoring Trinomials - Two Sums and Two Differences

Many trinomials are the product of two binomials. Factor a trinomial to create two binomials.

Given trinomial: x2 + 5x + 6

Step 1: Look for any Like terms to factor out! If there are not any continue to Step 2.

Step 2: Write the factors of x2: (x )(x )

Step 3: List all factors of the last term. 1,6 2,3

Step 4: Choose the factors whose sum equals the 2nd term. 2 + 3 = 5

Step 5: Put factors into the parentheses. (x + 2)(x + 3)

Step 6: Check your answer by multiplying your binomial pair (double distribute)

Guided Practice: Two Sums

1) x2 + 7x + 10 2) x2 + 10x + 16 3) x2 + 8x + 7 4) x2 + 4x + 4

(x )(x )

Two Differences

5) x2 - 8x + 15 6) x2 - 10x + 16 7) x2 - 5x + 4 8) x2 - 7x + 10

(x )(x )

Mixed

9) x2 + 6x + 9 10) x2 - 8x + 16 11) x2 + 4x + 4 12) x2 - 10x + 25

(x )(x )

Independent Practice: Factor each trinomial into a binomial pair:

1) x2 + 8x + 12 2) x2 - 12x + 35 3) x2 + 6x + 8 4) x2 - 9x + 20

(x )(x )

5) x2 + 8x + 16 6) x2 + 4x + 3 7) x2 - 12x + 36 8) x2 - 11x + 30

9) x2 + 14x + 49 10) x2 - 9x + 14 11) x2 + 9x + 18 12) x2 - 5x + 6

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Lesson 7 – Homework

Factor each trinomial into a binomial pair:

1) x2 + 5x + 6 2) x2 - 16x + 15 3) x2 + 8x + 12 4) x2 - 6x + 9

5) x2 + 11x + 24 6) x2 - 4x + 3 7) x2 + 6x + 8 8) x2 - 12x + 11

9) x2 + 2x + 1 10) x2 - 7x + 12 11) x2 + 8x + 7 12) x2 - 9x + 18

13) x2 + 7x + 10 14) - 7x + 10 15) x2 + 12x + 20 16) x2 - 11x + 18

Factor Out Like Terms

17) 2x + 6 18) 12x2 - 8 19) 7x2 + 21x 20) 3x5 - 15x4 + 6x2

21) 6c3 – 12c2 + 3c 22) 3a + 6a2 + 15a3 23) 12a2 + 20ab 24) 3x – 3y

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Lesson 8

Factor Trinomials One Sum - One Difference

Many trinomials are the product of two binomials. This is how you factor a trinomial.

x2 - 2x - 8

Step 1: Look for any Like terms to factor out! If there are not any continue to Step 2.

Step 2: Write: (x )(x )

Step 3: List all factors of the last number. -1, 8 -2, 4 -8, 1 -4, 2

Step 4: Choose the factors whose sum equals the 2nd term.

-1 + 8 = 7 -2 + 4= 2 -8 + 1 = -7 -4 + 2 = -2

Step 5: Put factors into the parentheses. (x - 4)(x + 2)

Step 6: Check your answer by multiplying your binomial pair (double distribute)

Guided Practice: Factor each trinomial into a binomial pair

1) x2 + 4x - 12 2) x2 - 2x - 15 3) x2 + 4x - 21 4) x2 + 5x - 6

(x )(x )

5) x2 - 2x - 24 6) x2 + 5x - 14 7) x2 - x - 6 8) x2 - 6x + 8

(x )(x )

Independent Practice: Factor each trinomial into a binomial pair

1) x2 + 7x - 18 2) x2 - x - 56 3) x2 + 11x + 30 4) x2 + x - 30

(x )(x )

5) x2 - 25x + 24 6) x2 + 3x - 10 7) x2 - 2x - 35 8) x2 - 16x - 17

(x )(x )

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Lesson 8 – Homework

Factor each trinomial into a binomial pair:

1) x2 + 4x - 12 2) x2 - 3x - 10 3) x2 + 5x - 24 4) x2 - 8x - 20

5) x2 + 2x - 15 6) x2 - 2x - 8 7) x2 + 8x - 33 8) x2 - 10x - 11

9) x2 + 6x - 16 10) x2 + 9x + 18 11) x2 + 10x - 24 12) x2 - 8x + 7

13) x2 + 10x - 39 14) x2 - 6x - 16 15) x2 + 9x - 22 16) x2 - 14x - 15

Review Work:

Multiply:

17) 8x(3x2 - x + 2) 18) (3x + 2)(2x - 3) 19) 6(7x - 3) 20) (x - 4)(x + 4)

Factor Out Like Terms:

21) 64x2 + 16x 22) 20x2y2 + 12xy 23) 4x2 + 42 24) 6x4 + 4x3+ 10x2

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Lesson 9

Factoring Trinomials Mixed Practice

Factor each trinomial into a binomial pair:

1) x2 + 3x - 18 2) x2 + 10x + 9 3) x2 - 9x + 20 4) x2 - 4x - 21

5) x2 + 10x + 25 6) x2 - 8x + 12 7) x2 + 8x - 33 8) x2 + 4x - 5

9) x2 - 4x + 4 10) x2 + 10x - 11 11) x2 - 10x - 24 12) x2 + 5x + 6

13) x2 + 2x - 3 14) x2 - 2x + 1 15) x2 - 3x - 4 16) x2 + 9x + 14

17) x2 + 3x - 10 18) x2 + 14x + 24 19) x2 - 7x - 18 20) x2 - 5x + 6

21) x2 + 6x - 7 22) x2 - 8x + 15 23) x2 - 3x - 28 24) x2 + 4x + 4

25) x2 + 2x - 35 26) x2 - 14x + 13 27) x2 - x - 6 28) x2 + 20x + 19

Factor out like terms:

29) 4x2 + 20x 30) 10x2y2 + 6xy 31) 36x12 + 42x10 32) 9x2 + 12x – 6

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Unit 14 Review

Lesson 1: Review Polynomials

Tell whether each is a monomial, binomial, or trinomial.

1) 6x + 8 2) 4x2y 3) x2 + 5x - 6

Simplify:

4) 4x + 11 – 3x + 4 – 6x 5) (2x – 14) + (13x – 5) 6) )26()1139( 22 ++−−+− xxxx

7) (x7)(x) 8) (9x2)(-2x5) 9) (4a2b5)(3a4b2)

Lessons 2 and 3: Find the product using either method.

10) (x – 4)(x + 3) 11) (x + 6)2

12) (5x + 2)( x – 3)

13) (2x – 1)2 14) (x + 2)(x – 2) 15) (8x – 6)(2x + 2)

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Lesson 4: Multiplying a Binomial by a Polynomial

Double Distribute using the Diagram Method: Double Distribute using either method:

16) (x – 3)(x2 – 3x + 4) 17) (3x + 1)(4x2 – 2y + 5)

Lesson 5: Find Greatest Common Factor

Find the GCF of the following:

18) 27x2 – 9x 19) 12x + 15 20) 10xy + 8xz

Lesson 6: Factor Out Like Terms

Factor:

21) 12x + 28 22) 5x – 15 23) 3x2 + 33x 24) 9y2 + 3y

25) If one factor of 16y2 + 12y is 4y, what is the other factor?

Lesson 7 and 8: Factor Trinomials

26) x2 + 5x + 6 27) x2 + 4x – 12 28) If one factor of x2 – 9x + 20 is (x – 5) what is

the other factor?

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Review Work:

29) 80 30) 8x0 31) (8x)0 32) 16(xyz)0

33) What is 2.7 x 105 written in standard form? 34) What is 5.63 x 10-4 written in standard form?

35) Is this a function? {(5, 6), (4, 9), (5, 0), (7, 1)} 36) Evaluate 6 + xy2 if x = 5 and y = 2:

37) What is the equation of a line with a slope of 3 and a y-intercept of -10?

38) What is the rate of change for a line passing through the following points? (2, 5) and (5, 11)

Draw any line with the following slopes:

39) Positive Slope 40) Negative Slope 41) Zero Slope 42) Undefined Slope

43) Will these angle measurements form a triangle? 10°, 50°, and 120°

44) What is the complement of 40°? 45) What is the supplement of 70°?

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46) State two angles that are:

a) Corresponding angles:________________________

b) Alternate Interior angles:______________________

c) Alternate Exterior angles:_____________________

d) Vertical angles:_____________________________

e) Supplementary angles:________________________

Solve the following systems:

47) 4x + 3y = 5 48) 2x + 3y = 4 49) 7x + 5y = 10

-4x – 3y = -5 5x – 3y = 10 -7x – 5y = 20

5 6

7 8

9 10

11 12