Algebra 2 Unit 3 - Polynomials Packet - Dam

UNIT 3:
Polynomial Functions

Algebra 2

Dr. Dam – [email protected]

Period:
Name:
ID:

SUCCESS CRITERIA 

 

I can…  factor or divide a polynomial by a linear binomial in order to retrieve the remainder. 

 

I can explain…  how to graph a polynomial function using end behavior and roots. 

 

I can justify…  the number, multiplicity, and types of roots of a polynomial function. 

 

I can model… a real-life scenario using a polynomial function. 

 

  TABLE OF CONTENTS   

LESSON  TITLE  PG # 
3.1  Polynomials (Key Terms & Key Features)  3-7 
3.2  Adding, Subtracting, and Multiplying Polynomials  8-10 
3.3  Dividing Polynomials  11-12 
3.4  Remainder Theorem + Factor Theorem  13-14 
3.5  Factoring + Rational Roots Theorem  15-18 
3.6  Graphing Polynomials  19-22 

  2 

Lesson 3.1 - Polynomials (Key Terms & Key Features)  N​ OT a Polynomial Function 
P​ olynomial Function 
g(x) = − 3 x3 + 7 −3
f (x) = 2x2 − 3x4 + 6x + 1 7 x

Key Terms: 
Polynomial 
Polynomial Function 
Standard Form 
Degree 
Type 
Leading Coefficient 
Leading Term 
Constant Term 

3 

Key Features: 

Domain 
Range 
x-Intercepts
y-Intercepts
Relative Maximum 
Relative Minimum 
Increasing 
Decreasing 
End Behavior 

Key Features: 

Domain 
Range 
x-Intercepts
y-Intercepts
Relative Maximum 
Relative Minimum 
Increasing 
Decreasing 
End Behavior 

4 

End Behavior​ (Degree & Leading Coefficient)

Describe the end behavior.
1) f (x) = 2x2 − 3x4 + 6x + 1

2) g(x) = 4 x + 6x2 + 3x5 − 3x3 − 2
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Polynomials (Key Terms) Review 

Riddle: WHAT DO YOU GET WHEN YOU CROSS AN EAR OF CORN WITH A SPIDER?

Write the polynomial function in standard form and then state its degree, type, and leading coefficient.
Write the letter of each answer in the box for each question number to find the answer to the riddle.

1) f (x) = 3 x3 − 2x + x4
4

2) f (x) = 12 − x + 2x2 − 4x

3) f (x) = 3x2 − x3 + 7x − 3

4) f (x) = √4x2 − 8

5) f (x) = 5 − 2 x3 + 6x − x2
5

√6) f(x) = 1 x + 10
4

7) f (x) = − 3x2 + x − x2 − 6

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Matching Activity 

1. y = − 3x + 2 5. y = − x6 − 7x3 + 2x + 15 9. y= 1 x6 − 7x3 + 2x
2. y = − x5 + 2x2 + 3 3
3. y = − x2 + 6x
4. y = x5 + 2x + 1 6. y = x4 − 2x3 − 3x2 + 4x 10. y = 2x3 + 4x2

7. y= 1 x2 − 4 11. y = − 2x3 − 2x2
2

8. y = 2x − 3 12. y = − 4x4 + 2x3 + 3x2 − 7x

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Lesson 3.2 - Adding, Subtracting, and Multiplying Polynomials 

Find the sum.
1) (10x4 + 3x2 − 5x + 4) + (7x5 − 5x4 + 2x − 9)

2) (5x4 + 3x2 − 6x − 10) + (2x3 − 7x2 + 6x + 1)

Find the difference.
3) (10x4 − 4x3 − 7x2 + 5x + 9) − (2x4 − 5x3 − 4x2 + 9x + 3)

4) (7x5 + 4x3 − 2x2 + 12x + 5) − (6x4 − 9x3 + x2 − 3)

8 

Find the product.
5) (8x2 − 3x + 1)(− 3x + 2)

6) (x + 3)(2x + 1)(2x − 3)

7) (y + 4)2

 
9 

Special Product Patterns 

Complete the table by finding each of the products. Make sure to show your work below the table.
In the last column, write an example of each special product.

Sum and Difference (a + b)(a − b) =

Square of a Binomial (a + b)2 =
(a − b)2 =

Cube of a Binomial (a + b)3 =
(a − b)3 =

Work:

10 

Lesson 3.3 - Dividing Polynomials 
 
 
Polynomial Long Division 

1) (x2 + x + 12) ÷ (x − 5)

2) (x3 + x2 − 9x − 6) ÷ (x2 − 9)

3) (x3 + 3x2 − 4x − 6) ÷ (x2 − 4)

 
 
 
 

 
11 

Synthetic Division 
When can synthetic division be used? 

For which problems can we use synthetic division?

(2x4 − 3x2 + 7x) ÷ (x − 7) (2x4 − 3x2 + 7x) ÷ (x2 − 7) (2x4 − 3x2 + 7x) ÷ (3x − 7)  

Dividing the following polynomials using Synthetic Division.
1) (3x2 − 2x + 6) ÷ (x − 1)

2) x3 − 2x + 6) ÷ (x + 3)

3) x3 + 5x3 − 6x2 − 11x + 14) ÷ (x + 4)

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Lesson 3.4 - Remainder Theorem + Factor Theorem 
Remainder Theorem 

If f (x) = 2x6 + 3x5 − 1 , then what is the remainder when f (x) is divided by x + 1 ?
Use synthetic division to evaluate the function for the indicated value of x .
1) f (x) = x3 + x2 − 4x + 3; x = − 1

2) f (x) = − x3 − x2 − 5; x = 3

3) f (x) = x4 + 5x2 − 8x + 1; find f (4)

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Factor Theorem 
 
 
 
 

 
 

 

Keep in mind, the following questions are asking the same thing: 
Is x + 2 a factor of x2 + 6x − 10 ? 
Is − 2 a root of x2 + 6x − 10 ? 
Is − 2 a solution of x2 + 6x − 10 ? 

1) Determine whether x + 5 is a factor of the polynomial function f (x) = 3x3 + 7x2 − 8x − 5 .

2) What is the value of k such that x − 6 is a factor of f (x) = 3x3 − 17x2 − kx + 18 ?

3) What is the value of k such that x + 1 is a factor of f (x) = x3 − kx − 12 ?

4) Determine whether 7 is a root of 8x3 + 2x2 = 49 .
4

 
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Lesson 3.5 - Factoring + Rational Roots Theorem 
 
 
Recall all the nicknames for roots from Unit 2? 

_____________________________________________________________________

 

Recall the F​ actoring Thinking Process:​  

1. GCF 

2. Binomials​ (N​ ote: Sum of squares cannot be factored unless there is a GCF)​  

a. Difference of squares: a2 − b2 =  

b. Difference of cubes: a3 − b3 =  

c. Sum of cubes: a3 + b3 =  

3. Trinomials 

a. Perfect square trinomial: a2 + 2ab + b2 = a2 − 2ab + b2 =  

b. x2 + bx + c (a = 1)  

c. ax2 + bx + c (a =/ 1)  

4. Four terms →​ grouping 

 

 

Which ones have we learned/reviewed so far? 

 

In this unit, we will focus on ​2b,​ 2​ c​, and ​4​. 

 

 

 

 

 

 
 
 

 
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Factoring Polynomials  
 

Sum of Cubes  Grouping 
 

a3 + b3 =

Ex 1)​ 8m3 + 125n3 Ex 1) 2a3 − a2b + 10a − 5b

Ex 2)​ 64 + 27d3

a3 − b3 =   Difference of Cubes  Ex 2) 48v3 − 24v2 − 56v + 28
 
 
Ex 1)​ 216x3 − 27   16 
16v3 − 54w3  
 
 
 
 
 

Ex 2)​

 
 
 
 
 
 

 

 

 
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Sometimes, a polynomial is not factorable, even by the grouping method. 
In these situations, _​ _________________________________________________________ 

Rational Root Theorem 

*This is only for RATIONAL ROOTS (roots that are integers/fractions).
Complex Conjugates Theorem 

1) What are all the possible rational roots of h(x) = x2 − 2 ?

Use the Rational Root Theorem to find all the possible rational roots first.
Then use synthetic division to find ALL the roots of each polynomial.

2) f (x) = 2x3 − 3x2 − 11x + 6 3) g(x) = 3x3 − 4x2 − 17x + 6

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Lesson 3.6 - Graphing Polynomials  Graphs  x-intercepts  # of x-int 
     
 
 
Functions    
1. y = x3
 
2. y = (x + 2)3     

3. y = (x − 3)(x + 1)2    
  
4. y = − (x − 3)2(x + 1)  
 
5. y = x(x − 3)(x + 2)2     

6. y = − (x + 4)2(x − 1)2(x + 2)    
  
 
   

 
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What happens in the middle? 
How to draw repeated roots: 

➢ A factor that is raised to the first power produces a _​ ____________________________________________​ .​  
➢ A factor that is squared produces a _​ _______________________________________________________​ ​. 
➢ A factor that is cubed produces a _​ _________________________________________________________​ .​  

Some examples: 
Notice the patterns in the degree (even/odd), leading coefficient (+/-), roots/zeros, and multiplicities. 

 

y = x(x + 3)(x − 2) y = − (x + 3)3(x − 1) y = = x2(x − 3)3

 
 
 

 

Steps to ​sketch​ polynomial functions in factored form: 
1. Determine the end behavior of the function using the function’s leading term. 
2. Find the zeros and identify their multiplicities. 
3. Sketch the graph based on the results of your findings in Step 1 & 2. 

 
20 

Sketch the graph of each polynomial function. 4) y = − (x + 2)2(x − 4)3
1) y = 2x(x − 3)2(x + 4)

2) y = − (x + 1)2(x + 3) 5) y = x2(x + 1)(x − 5)2

3) y = (x − 5)3(x + 2) 6) y = − (x + 3)3(x − 5)

 
21 

 

 

Steps to ​graph​ polynomial functions: 

1. Determine the end behavior of the function using the function’s leading term. 

2. Find the zeros and identify their multiplicities. 

3. Find the y-intercept by substituting x = 0 . 

4. Graph the polynomial based on the results of your findings in Step 1, 2 ,3. 

 

 
 

1) f (x) = (x − 4)(x − 1)(x + 3)

Leading Term End Behavior

Zeros
Multiplicity

y-intercept

 
22 

2) g(x) = (x + 1)(x − 2)2(x − 4)

Leading Term End Behavior

Zeros
Multiplicity

y-intercept

3) h(x) = − x3(2x − 3)

Leading Term End Behavior

Zeros
Multiplicity

y-intercept

 
23 

4) j(x) = 1 (x − 2)3(x + 3)2
10

Leading Term End Behavior

Zeros
Multiplicity

y-intercept

5) k(x) = x3 + 2x2 − 8x
Factor:

Leading Term End Behavior

Zeros
Multiplicity

y-intercept

 
24 

6) m(x) = x3 + 2x2 − 36x − 72
Factor:

Leading Term End Behavior

Zeros
Multiplicity

y-intercept

 
 
 
 
 

 
25 

Factoring Polynomials Review 

Activity 1 https://teacher.desmos.com/activitybuilder/custom/56b90984064eaf004334ee69

Activity 2 Some cubic equations have three distinct solutions. Others have repeated solutions. 

Directions:
1. Match each cubic polynomial equation with its correct graph.
2. Then, solve each equation to find its solutions.
3. For equations with repeated solutions, describe the behavior of the function at each zero.

_____ 1​ .​ x3 − 6x2 + 12x − 8 = 0 _____ ​4.​ x3 + x2 − 2x = 0

_____ 2​ .​ x3 + 3x2 + 3x + 1 = 0 _____ ​5.​ x3 − 3x − 2 = 0

_____ 3​ .​ x3 − 3x + 2 = 0 _____ 6​ .​ x3 − 3x2 + 2x = 0

A. C. E.
B. D. F.

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