Unit 3 Polynomials

Test date: _____________________

Name: ___________________________________________________________________________________

By the end of this unit, you will be able to…

Identify key components of polynomial functions Graph a polynomial function given the degree, zeros, and end behavior Analyze a graph based on turning points, zeros, relative maxima and minima, and end behavior Solve polynomial equations by factoring Write a polynomial function in standard form given the real and imaginary zeros Simplify an expression using exponent rules Add, subtract, multiply, and divide polynomials Evaluate functions using synthetic substitution (the Remainder Theorem) Find the zeros of a function using the graph, then use synthetic division to find remaining zeros Describe the relationship of zeros, factors, roots, and x-intercepts of polynomials

2

Table of Contents

5.1 Exponent Rules ........................................................................................................................................................................... 3

Simplifying Polynomial Expressions ..................................................................................................................................... 5

Multiplying Polynomials ............................................................................................................................................................ 5

Using the Distributive Property .............................................................................................................................................. 6

5.2 Polynomial Long Division ....................................................................................................................................................... 7

5.2 Synthetic Division ...................................................................................................................................................................... 8

5.3 Power Functions Lab .............................................................................................................................................................. 10

5.3 Polynomial Functions ............................................................................................................................................................. 12

Evaluating Polynomial Functions ......................................................................................................................................... 12

5.3 Graphing Polynomials and Using End Behavior .......................................................................................................... 14

5.4 Analyzing Graphs of Polynomial Functions ................................................................................................................... 16

5.5 Factoring Polynomials ........................................................................................................................................................... 19

Sum and Difference of Cubes ................................................................................................................................................. 19

Factoring Techniques Summary ........................................................................................................................................... 20

Factoring by Grouping .............................................................................................................................................................. 21

5.6 Remainder and Factor Theorems ...................................................................................................................................... 23

Remainder Theorem ................................................................................................................................................................. 23

Factor Theorem ........................................................................................................................................................................... 24

Finding Zeros without a Starting Point .............................................................................................................................. 25

5.7 Roots and Zeros ........................................................................................................................................................................ 26

Graphs of Polynomial Functions ........................................................................................................................................... 28

5.7 Graphing Polynomial Inequalities ..................................................................................................................................... 30

3

5.1 Exponent Rules Directions: Without using the textbook, simplify the following expressions by filling in the missing values.

Properties of Exponents Describe the short cut for each property below:

1) Product of Powers

𝑥3 ∙ 𝑥4 = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 = 𝑥___ _________________________________

2) Quotient of Powers 𝑦6

𝑦4=

𝑦∙𝑦∙𝑦∙𝑦∙𝑦∙𝑦

___∙___∙___∙___= ___ ∙ ___ = 𝑦2 _________________________________

3) Negative Exponents

𝑏2

𝑏5= 𝑏2−5 = 𝑏___ _________________________________

𝑏2

𝑏5=

___ ∙ ___

𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏=

1

𝑏 ∙ 𝑏 ∙ 𝑏=

1

𝑏3

4) Power of a Power

(𝑐2)3 = 𝑐2 ∙ ___ ∙ ___ = 𝑐 ∙ ___ ∙ ___ ∙ ___ ∙ ___ ∙ ___ = 𝑐___ ____________________________

5) Power of a Product

(7𝑎2)3 = 7𝑎2 ∙ ____ ∙ ____ __________________________________

= 7 ∙ 7 ∙ 7 ∙ ___ ∙ ___ ∙ ___ ∙ ___ ∙ ___ ∙ ___

= 7___ ∙ 𝑎___

6) Power of a Quotient __________________________________

(8

𝑧)

4

=8

𝑧∙ ∙ ∙ =

8___

𝑧___

7) Zero Power __________________________________

𝑝5

𝑝5= 𝑝___−___ = 𝑝___= ____

𝑝5

𝑝5= ____

4

Properties of Exponents: For any real numbers a and b, integers m and n.

Property Definition Example

Product of Powers Property

Power of a Power Property

Power of a Product

Negative Exponent Property

Zero Exponent Property

Quotient of Powers Property

Power of a Quotient

Practice: Simplify.

1. 4−4(46)−2(4) 2. 2−3𝑎5𝑏7𝑐−2

3−2𝑎8𝑏7𝑐

3. (−2

3)

−2

4. 2𝑥3

𝑦−2∙ (

2𝑥−1𝑦2

2)

3

5. (𝑥−3𝑦5

2𝑥2𝑦2)−4

6. (−2𝑎2𝑏5

3𝑐0𝑎2)2

5

Monomial: ____________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ Polynomial: ___________________________________________________________________________________________________________ Degree: _______________________________________________________________________________________________________________ Determine whether the following are polynomials. If it is a polynomial, state the degree.

1. −16𝑝5 +3

4𝑝2𝑞7 2. 𝑥2 − 3𝑥−1 + 7 3.

1

2𝑎2𝑏3 + 3𝑐5

Simplifying Polynomial Expressions

1. (2𝑎3 + 5𝑎 − 7) − (𝑎3 − 3𝑎 + 2) 2. (4𝑥2 − 9𝑥 + 3) + (−2𝑥2 − 5𝑥 − 6)

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

Multiplying Polynomials

1. (𝑎2 + 3𝑎 − 4)(𝑎 + 2) 2. (𝑥2 + 3𝑥 − 2)(𝑥 + 4)

3. The shipping container is a rectangular prism. Write a polynomial that represents the volume of the container.

6

4. (𝑎2 + 3𝑎 − 4)(𝑎2 + 2) + 3𝑎4(3𝑎2 − 4𝑎) 5. (𝑦 − 1)3

Using the Distributive Property

1. – 𝑦(4𝑦2 + 2𝑦 − 3) 2. – 𝑥(3𝑥3 − 2𝑥 + 5) Applications

1. You plan to hire a gardener to work for 15 hours to plant your veggies. There are two available options, but neither one can work all 15 hours. Gardener 1 charges $14 per hour and Gardener 2 charges $17 per hour. If the first one is only available for x hours, write an expression to represent the total cost.

2. The U.S. Dept. of Transportation limits the time a truck driver can work to 10 hours. For the first shift, Tom drives at a speed of 60 mph and for the 2nd part, he drives at a speed of 70 mph. If the truck driver limits the time traveling 60 mph to x hours, write a polynomial to represent the total distance driven.

3. Olivia has $1300 to invest in a government bond that has an annual interest rate of 2.2% and a savings account that pays 1.9% per year. Write a polynomial for the interest she will earn in one year if she invests x dollars in the government bond.

7

5.2 Polynomial Long Division Warm Up: Simplify. Dividing a Polynomial by a Monomial

1. 5𝑎2𝑏−15𝑎𝑏3+10𝑎3𝑏4

5𝑎𝑏 2.

3𝑥2𝑦+6𝑥5𝑦2−9𝑥7𝑦3

3𝑥2𝑦

Long Division

1. (𝑥2 − 2𝑥 − 15) ÷ (𝑥 − 5) 2. (𝑥2 + 5𝑥 + 6) ÷ (𝑥 + 3)

3. (𝑎2 − 5𝑎 + 3)(2 − 𝑎)−1 4. (𝑥2 − 𝑥 − 7)(𝑥 − 3)−1

5. (6𝑦2 − 5𝑦 − 15)(2𝑦 + 3)−1 6. 4𝑥2−2𝑥+6

2𝑥−3

1. 𝑏2 ∙ 𝑏5 ∙ 𝑏3 2. 6𝑎𝑐4

−3𝑎2𝑐2

3. (10𝑎2 − 6𝑎𝑏 + 𝑏2) − (5𝑎2 − 2𝑏2) 4. 7𝑤(2𝑤2 + 8𝑤 − 5) 5. State the degree of 6𝑥𝑦2 − 12𝑥3𝑦2 + 𝑦4 − 26. 6. Find the product of 3𝑦(2𝑦2 − 1)(𝑦 + 4).

8

5.2 Synthetic Division

Synthetic division a procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x – r

Use synthetic division to find (2𝑥3 – 5𝑥2 + 5x – 2) ÷ (x – 1).

Step 1 Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients.

2𝑥3 – 5𝑥2 + 5x – 2

2 –5 5 –2

Step 2 Write the constant r of the divisor x – r to the left, In this case, r = 1. Bring down the first coefficient, 2, as shown.

1 2 –5 5 –2

2

Step 3 Multiply the first coefficient by r, 1 ⋅ 2 = 2. Write their product under the second coefficient. Then add the product and the second coefficient: –5 + 2 = – 3.

1 2 –5 5 –2

2

2 –3

Step 4 Multiply the sum, –3, by r: –3 ⋅ 1 = –3. Write the product under the next coefficient and add: 5 + (–3) = 2.

1 2 –5 5 –2

2 -3

2 –3 2

Step 5 Multiply the sum, 2, by r: 2 _ 1 = 2. Write the product under the next coefficient and add: –2 + 2 = 0. The remainder is 0.

1 2 –5 5 –2

2 -3 2

2 –3 2 0

Thus, (2𝑥3 – 5𝑥2+ 5x – 2) ÷ (x – 1) = 2𝑥2 – 3x + 2. Use synthetic division to find the following.

1. (x3 – 4x2 + 6x – 4) ÷ (x – 2) 2. (4x3 + x2 + 8x + 7) ÷ (x + 1)

3. (𝑝3 − 6) ÷ (𝑝 − 1) 4. (2x3 +3x2 - 4x + 15) ÷ (x + 3)

Use 0 for missing coefficients!

9

5. (4x4 +2x2 - 4x + 12) ÷ (x + 2) 6. (6𝑥4 − 8𝑥3 + 12𝑥 − 14) ÷ (𝑥 − 2)

7. (𝑥2 + 8𝑥 + 7) ÷ (𝑥 + 1) 8. (4y3 – 6y2 + 4y – 1) ÷ (2y – 1)

9. (8y3 – 12y2 + 4y + 10) ÷ (2y + 1) 10. (3𝑥4 − 5𝑥3 + 𝑥2 + 7𝑥) ÷ (3𝑥 + 1) Applications

11. Mr. Collins has his class working with bases and polynomials. He wrote on the board that the number 1111 in base B has the value 𝐵3 + 𝐵2 + 𝐵 + 1. The class was then given the following questions to answer.

a. The number 11 in base B has the value 𝐵 + 1. What is 1111 (in base B) divided by 11 (in base B)?

b. The number 111 in base B has the value 𝐵2 + 𝐵 + 1. What is 1111 (in base B) divided by 111 (in base B)?

Divisor with 1st coefficient other than 1? Divide everything by that coefficient.

10

5.3 Power Functions Lab A power function is any function of the form 𝑓(𝑥) = 𝑎𝑥𝑛 , where 𝑎 and 𝑛 are nonzero constant real numbers. A power function in which 𝑛 is a positive integer is called a monomial function. Activity 1: 𝒚 = 𝒂𝒙𝟒

1. Graph the function 𝑦 = 2𝑥4. 2. Analyze the graph.

Analyze the Results:

1. What assumptions can you make about the graph of 𝑓(𝑥) = 2𝑥4 as x becomes more positive or more negative? Use the [TABLE] feature to confirm your assumptions.

2. State the domain and range of 𝑓(𝑥) = 2𝑥4. Compare these values with the domain and range of 𝑔(𝑥) = 𝑥2.

3. What other characteristics does the graph of 𝑓(𝑥) = 2𝑥4 share with the graph of 𝑔(𝑥) = 𝑥2? Activity 2: 𝒚 = 𝒂𝒙𝟓

1. Graph the function 𝑦 = 3𝑥5 (sketch at right). 2. Analyze the graph.

Analyze the Results:

1. What assumptions can you make about the graph of 𝑓(𝑥) = 3𝑥5 as 𝑥 becomes more positive or more negative? Use the [TABLE] feature to confirm your assumption.

2. State the domain and range of 𝑓(𝑥) = 3𝑥5. Compare these values with the domain and range of 𝑔(𝑥) = 𝑥3.

3. What other characteristics does the graph of 𝑓(𝑥) = 3𝑥5 share with the graph of 𝑔(𝑥) = 𝑥3?

4. Compare the characteristics of the graph ℎ(𝑥) = −3𝑥5 with the graph of 𝑓(𝑥) = 3𝑥5. What conclusions can you make about 𝑓(𝑥) = 𝑎𝑥5 when 𝑎 is positive and when 𝑎 is negative?

11

Activity 3: 𝒚 = 𝒙𝒏, where 𝒏 is even.

1. Graph 𝑓(𝑥) = 𝑥2, 𝑔(𝑥) = 𝑥4, and ℎ(𝑥) = 𝑥6 on the same screen (sketch at right).

2. Analyze the graphs. Analyze the Results:

1. What assumptions can you make about the graphs of 𝑎(𝑥) = 𝑥8, 𝑏(𝑥) = 𝑥10, and so on?

2. Identify the common characteristics of the graphs of power functions in which the power is an even number.

3. Graph 𝑓(𝑥) = 𝑥3, 𝑔(𝑥) = 𝑥5, and ℎ(𝑥) = 𝑥7 on the same graph (sketch at right).

4. What assumptions can you make about the graphs of 𝑎(𝑥) = 𝑥9, 𝑏(𝑥) = 𝑥11, and so on?

5. Identify the common characteristics of the graphs of power functions in which the power is an odd number.

6. Graph 𝑓(𝑥) = 𝑥4 and 𝑔(𝑥) = −𝑥4 on the same graph (sketch at right).

7. Graph 𝑓(𝑥) = 𝑥5 and 𝑔(𝑥) = −𝑥5 on the same graph (sketch at right).

8. What assumptions can you make about the effects of a negative value of 𝑎 in 𝑓(𝑥) = 𝑎𝑥𝑛 when 𝑛 is even? When 𝑛 is odd?

12

5.3 Polynomial Functions

Polynomial in One Variable

A polynomial of degree n, in one variable x is an expression of the form

𝑎𝑛𝑥𝑛 + 𝑎𝑛 − 1𝑥𝑛 − 1 + … + 𝑎2𝑥2 + 𝑎1x + 𝑎0, where 𝑎𝑛 is not ____, the coefficients 𝑎𝑛 − 1, 𝑎2, 𝑎3, …,

𝑎0 are _________________, and n represents a _______________________________.

When is a polynomial considered to be in standard form? ___________________________________________________ What is the degree of a polynomial? _____________________________________________________________________ What is the leading coefficient? _______________________________________________________________________ Polynomial Vocabulary

Polynomial Example Expression Degree of your Example L.C. of your Example Constant Linear Quadratic Cubic Example: What are the degree and leading coefficient of 3𝒙𝟐 – 2𝒙𝟒 – 7 + 𝒙𝟑? 1. Rewrite the expression so the powers of x are in decreasing order. ________________________________________

2. What is the degree?___________ 3. What is the leading coefficient?_______________

Exercises: State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why.

6. 3𝑥4 + 6𝑥3 – 𝑥2 + 12 7. 100 – 5𝑥3 + 10 𝑥7 8. 4𝑥6 + 6𝑥4 + 8𝑥8 – 10𝑥2 + 20

9. 4𝑥2 – 3xy + 16𝑦2 10. 8𝑥3 – 9𝑥5 + 4𝑥2 – 36 11. 𝑥2

18 –

𝑥6

25 +

𝑥3

36 –

1

72

Evaluating Polynomial Functions

Application: The volume of air in the lungs during a 5-second respiratory cycle can be modeled by

𝑣(𝑡) = −0.037𝑡3 + 0.152𝑡2 + 0.173𝑡, where v is the volume in liters and t is the time in seconds. This model is an example of a polynomial function. Find the volume of air in the lungs 1.5 seconds into the respiratory cycle.

13

Example: Find f(–5) if f(x) = 𝒙𝟑 + 2𝒙𝟐 – 10x + 20.

f(x) = 𝑥3 + 2𝑥2 – 10x + 20 Original function

f(–5) = ( )3 + 2( )2 – 10( ) + 20 Replace x with –5.

= _____ + _____ + _____ + _____ Evaluate.

= ______ Simplify.

Find f(2) and f(–5) for each function. 12. f(x) = 𝑥2 – 9 13. f(x) = 4𝑥3 – 3𝑥2 + 2x – 1 14. f(x) = 9𝑥3 – 4𝑥2 + 5x + 7 You can evaluate polynomials using synthetic substitution as well. This is just like synthetic division from yesterday. The final term is what the function evaluates to. See #15 for an example! Find f(-2) for each function. 15. f(x) = -2x4 – 2x3 + 2x – 9 16. f(x) = -x5 – x4 + 3x3 + 2x2 – x + 1 17. f(x) = 3x3 – 2x2 + 4 Example: Find g(𝒂𝟐 – 1) if g(x) = 𝒙𝟐 + 3x – 4.

g(x) = 𝑥2 + 3x – 4 Original function

g(𝑎2 – 1) = (_______)2 + 3(_______) – 4 Replace x with 𝑎2 – 1.

= Evaluate.

= Simplify. If p(x) = 3𝒙𝟐 – 4 and r(x) = 2𝒙𝟐 – 5x + 1, find each value. 18. p(8a) 19. r(𝑎2) 20. 2r(x + 2) 21. p(𝑥2 – 1)

14

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

5.3 Graphing Polynomials and Using End Behavior 1. Graph f(x) = 3x – 4. Use a table of values. 2. What is the degree of this function?____ 3. As the x-values approach infinity (increase), describe what happens to f(x) values.

_________________________________________________________________ 4. As the x-values approach negative infinity, describe what happens to the f(x) values.

_________________________________________________________________ 5. Graph f(x) = -½x + 1. Use a table. 6. What is the degree of this function?____ 7. As the x-values approach infinity (increase), describe what happens to f(x) values.

_________________________________________________________________ 8. As the x-values approach negative infinity, describe what happens to f(x) values.

_________________________________________________________________ 9. Graph f(x) = x2 + x – 4. Use a table of values. 10. What is the degree of this function?____ 11. As the x-values approach infinity (increase), describe what happens to f(x) values.

_________________________________________________________________ 12. As the x-values approach negative infinity, describe what happens to f(x) values.

_________________________________________________________________ 13. Graph f(x) = -x2 + 1. Use a table of values. 14. What is the degree of this function?______ 15. As the x-values approach infinity (increase), describe what happens to f(x) values.

_________________________________________________________________ 16. As the x-values approach negative infinity, describe what happens to f(x) values.

_________________________________________________________________

x -2 -1 0 1 2

y

x -2 -1 0 1 2 y

X -2 -1 0 1 2 Y

X -2 -1 0 1 2 Y

15

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

17. Graph f(x) = 2x3 – x2 – 3x – 4. Use a table of values. 18. What is the degree of this function? 19. As the x-values approach infinity (increase), describe what happens to f(x) values. ______________________________________________________________________________________ 20. As the x-values approach negative infinity, describe what happens to f(x) values. ______________________________________________________________________________________

21. Graph f(x) = -2x3 – x2 + 3x – 4. Use a table of values. 22. What is the degree of this function? 23. As the x-values approach infinity (increase), describe what happens to f(x) values. ______________________________________________________________________________________ 24. As the x-values approach negative infinity, describe what happens to f(x) values. ______________________________________________________________________________________ 25. Graph f(x) = x4 + x – 4. Use a table of values. 26. What is the degree of this function? 27. As the x-values approach infinity (increase), describe what happens to f(x) values. ______________________________________________________________________________________ 28. As the x-values approach negative infinity, describe what happens to f(x) values. ______________________________________________________________________________________

29. Graph f(x) = -2x4 + x – 4. Use a table of values. 30. What is the degree of this function? 31. As the x-values approach infinity (increase), describe what happens to f(x) values. ______________________________________________________________________________________ 32. As the x-values approach negative infinity, describe what happens to f(x) values. ______________________________________________________________________________________

END BEHAVIOR GENERAL RULES -means get smaller (approaches negative infinity), +means get larger (approaches positive infinity)

Sign of Leading Coefficient Degree (Even or Odd) As x - , f (x) ______ As x + , f (x) ______

X -2 -1 0 1 2 Y

X -2 -1 0 1 2 Y

X -2 -1 0 1 2 Y

X -2 -1 0 1 2 Y

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

16

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

5.4 Analyzing Graphs of Polynomial Functions

1. Graph 𝑓(𝑥) =1

4(𝑥 + 2)(𝑥 − 1)2

Local maximum: _________________________________________________________ Local minimum: _________________________________________________________ Turning points: __________________________________________________________ ____________________________________________________________________________ Bounce point: ________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ The graph of every polynomial function of degree n has at most __________ turning points. If a polynomial function has ___________________________________, then its graph has ________________________________________________. Graphing Calculator Practice: Identify x-intercepts and local maxima and minima using [CALC] menu.

1. 𝑓(𝑥) = 𝑥3 − 3𝑥2 + 5 2. 𝑓(𝑥) = 𝑥3 + 3𝑥2 + 2

Applications:

1. The weight w, in pounds, of a patient during a 7-week illness is modeled by the function 𝑤(𝑛) = 0.1𝑛3 − 0.6𝑛2 + 110, where n is the number of weeks since the patient became ill. Graph the equation using the table provided. n 0 1 2 3 4 5 6 7 8

w(n)

2. Describe the end behavior and turning points.

3. What trends in the patient’s weight does the graph suggest?

4. Is it reasonable to assume the trend will continue indefinitely?

Polynomial Graph Must Include: -ALL zeros -ALL local extrema (max/min) -arrows showing end behavior

17

En

d B

eh

av

ior

Do

ma

in

an

d

Ra

ng

e

Re

lati

ve

Ma

x/

Min

(w

rite

th

e o

rder

ed

pai

rs)

Gra

ph

(

incl

ud

e al

l ro

ots

w

ith

ord

ered

pai

rs)

Fu

nct

ion

O

dd

/Ev

en

D

eg

ree

A

nd

Lis

t Z

ero

s

𝑓( 𝑥

)=𝑥

4−𝑥

3−

4𝑥

2+

1

𝑓( 𝑥

)=𝑥

3−

4𝑥

2+

2

𝑓( 𝑥

)=𝑥

5−

4𝑥

3+

3𝑥

2−

6

𝑓( 𝑥

)=

−𝑥

3+

2𝑥

2−

4

18

Sketch a graph of each function. Use the min/max and intersect options on your graphing calculator to answer the following. Describe the end behavior of each function. 1. For 1980 through 1996, the total exports (E) in billions of dollars of the US can be modeled by: E = -.131t3 + 5.033t2 – 23.2t + 233, where t is the number of years since 1980. In what year were the total exports $312.76 billion? 2. For 1983 through 1995, the amount of private donations (D) in millions of dollars allocated to education can be modeled by D = 1.78t3 – 6.02t2 + 752t + 6701, where t is the number of years since 1983. In what year was $14.3 billion of private donations allocated to education? 3. The average amount of oranges (in pounds) eaten per person each year in the US from 1991 to 1996 can be modeled by f(x) = .298x3 – 2.73x2 + 7.05x + 8.45, where x is the number of years since 1991. Graph, identify any local max/mins on the interval 0 < x < 5, and determine what meanings these points have. 4. The volume of a box that is created from a piece of cardboard that is 18 inches by 18 inches can be modeled by V = 4x3 – 72x2 + 324x, where x is the length of the cut from one of the sides. What is the length of the cut that will maximize the volume of the box? What is the max volume? 5. For 1987 through 1996, the sales (S) in millions of dollars of gym shoes can be modeled using the equation S = -.982t5 + 24.6t4 – 211t3 + 661t2 – 318t + 1520, where t is the number of years since 1987. Were there any years in which sales were about $2 billion? Explain. 6. For 1990 through 2000, the amount of money projected to be spent on television per person per year in the US can be modeled by S = -.213t3 + 3.96t2 + 10.2t + 366, where S is the amount spent, and t is the number of years since 1990. During which year was $455 spent per person on television. 7. The polynomial function 𝑃 = .134𝑡3 − 5.775𝑡2 + 70.426𝑡 + 481.945 models the number of points earned by the gold medal winner of the platform diving event in the summer Olympics, where t is the number of years since 1972. Graph the function and identify any turning points on the interval 0 ≤ 𝑡 ≤ 24. What real life meaning do these points have?

19

5.5 Factoring Polynomials

Sum and Difference of Cubes Factoring Technique General Case Sum of Two Cubes 𝑎3 + 𝑏3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2) Difference of Two Cubes 𝑎3 − 𝑏3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2) Examples: Factor the polynomial expression. If it is not factorable, write “prime”. 1. 𝑥3 + 27 2. 8𝑥3 + 27 3. −27𝑢3 − 125 4. 250𝑥4 + 128𝑥

5. 64𝑥3 − 1 6. 21𝑥9 + 2𝑦9 7. 343𝑚3 + 64𝑛3 8. 𝑥3 − 216𝑦3

Solve each equation by factoring. Then, graph the related function and label the real zeros on the graph.9. 𝑥3 + 27 = 0 10. 8𝑥3 − 1 = 0

3( ) 27f x x

3( ) 8 1f x x

20

11. 38 64 0x

12. 5 254 250 0x x Solve each equation by factoring.

13. 5 2216y y 14. 3𝑥7 = 81𝑥4

Factoring Techniques Summary Number of Terms Factoring Technique General Case Any number

Two

Three

Four or more

3( ) 8 64f x x

5 2( ) 54 250f x x x

21

Factoring by Grouping Factor the polynomial expression. If it is not factorable, write “prime”. 1. −2𝑥3 − 4𝑥2 − 3𝑥 − 6 3. 3𝑥3 − 2𝑥2 − 9𝑥 + 6

2. 2𝑥5 + 4𝑥4 − 4𝑥3 − 8𝑥2 4. −3𝑥3 + 7𝑥2 + 12𝑥 − 28

Solve the equation by factoring. Then, graph the related function and label the real zeros on the graph.5. 2𝑥3 − 3𝑥2 − 10𝑥 = −15 Rewrite the function in factored form. Find all zeros (including imaginary). Then, graph the function. Be sure to include all real zeros on the graph. Also, find and graph all local minima and local maxima. 6. Domain: _________________________

Range: __________________________

Zeros: ___________________________

Rel Min: _________________________

Rel Max: _________________________

End Behavior: _____________________

_____________________

3 2( ) 2 3 10 15f x x x x

3 2( ) 2 18 9f x x x x

Directions: 1. Separate pairs of terms with parentheses 2. Factor a GCF from each pair 3. The coefficients (GCFs) become a factor

22

Rewrite the function in factored form. Find all zeros (including imaginary). Then, graph the function. Be sure to include all real zeros on the graph. Also, find and graph all local minima and local maxima. 7. Domain: _________________________

Range: __________________________

Zeros: ___________________________

Rel Min: _________________________

Rel Max: _________________________

End Behavior: _____________________

_____________________

8. Domain: _________________________

Range: __________________________

Zeros: _______________________

Rel Min: _________________________

Rel Max: _________________________

End Behavior: _____________________

_____________________

9. Domain: _________________________

Range: __________________________

Zeros: ___________________________

Rel Min: _________________________

Rel Max: _________________________

End Behavior: _____________________

_____________________

3 2( ) 3 4 3 4f x x x x

3 2( ) 4 4f x x x x

3 2( ) 3 15 5f x x x x

23

5.6 Remainder and Factor Theorems Warm Up: Solve by factoring. Then, graph the related function.

1. 𝑥3 − 2𝑥2 − 9𝑥 + 18 = 0 𝑓(𝑥) = 𝑥3 − 2𝑥2 − 9𝑥 + 18 2. −2𝑥3 + 4𝑥2 + 3𝑥 − 6 = 0 𝑓(𝑥) = −2𝑥3 + 4𝑥2 + 3𝑥 − 6

Remainder Theorem If a polynomial 𝑃(𝑥) is divided by 𝑥 − 𝑟, the remainder is a constant 𝑃(𝑟), and 𝑃(𝑥) = 𝑄(𝑥) ∙ (𝑥 − 𝑟) + 𝑃(𝑟), where 𝑄(𝑥) is a polynomial with degree one less than 𝑃(𝑥). You can use the Remainder Theorem to evaluate functions with synthetic substitution. To find 𝑓(𝑥), use 𝑥 in the synthetic division algorithm. The remainder is the value of 𝑓(𝑥). Example: If 𝑓(𝑥) = 2𝑥4 − 5𝑥2 + 8𝑥 − 7, find 𝑓(6). Synthetic Substitution Direct Substitution

24

Factor Theorem The binomial 𝑥 − 𝑟 is a factor of the polynomial 𝑃(𝑥) if and only if ______________________________. Remainder of zero → _______________________________________ The Factor Theorem can be used to determine whether a binomial is a factor of a polynomial. It can also be used to find all the factors of a polynomial. Examples:

1. Determine whether 𝑥 − 3 is a factor of 𝑥3 + 4𝑥2 − 15𝑥 − 18. Then, find the remaining factors of the polynomial.

2. Determine whether 𝑥 + 2 is a factor of 𝑥3 + 8𝑥2 + 17𝑥 + 10. If so, completely factor the polynomial.

3. Find all zeros of 𝑓(𝑥) = 2𝑥3 + 11𝑥2 + 18𝑥 + 9, given that 𝑓(−3) = 0.

4. Factor and find the zeros of 𝑓(𝑥) = 3𝑥3 + 13𝑥2 + 2𝑥 − 8, given that 𝑓(−4) = 0.

5. Find all the zeros of 𝑓(𝑥) = 𝑥3 − 2𝑥2 − 9𝑥 + 18 given that one zero is 𝑥 = 2.

6. Find all zeros of 𝑓(𝑥) = 𝑥3 + 6𝑥2 + 3𝑥 − 10 given that one zero is 𝑥 = −5.

25

Warm Up:

1. Factor 𝑓(𝑥) = 𝑥3 − 11𝑥2 + 14𝑥 + 80, given 𝑓(8) = 0.

2. Find all zeros of 𝑓(𝑥) = 4𝑥3 + 9𝑥2 − 52𝑥 + 15 given that (𝑥 + 5) is a factor.

Finding Zeros without a Starting Point Examples:

1. 𝑓(𝑥) = 𝑥3 + 2𝑥2 − 11𝑥 − 12 2. 𝑓(𝑥) = 𝑥3 − 4𝑥2 − 11𝑥 + 30

3. 𝑓(𝑥) = 𝑥4 − 6𝑥3 + 9𝑥2 + 6𝑥 − 10 4. 𝑓(𝑥) = 𝑥3 + 7𝑥2 + 4𝑥 + 28

5. 𝑓(𝑥) = 3𝑥3 − 𝑥2 + 9𝑥 − 3 6. 𝑓(𝑥) = 𝑥4 − 2𝑥3 − 8𝑥2 − 32𝑥 − 384

7. 𝑓(𝑥) = 𝑥3 − 4𝑥2 − 15𝑥 + 68

26

5.7 Roots and Zeros 1. Factor to solve for all Zeros of 2. Use Synthetic Division to Solve for all Zeros of: f(x) = x3 + 5x2 + 2x + 10 g(x) = x4 – 18x2 + 12x + 80 Key Concept: The Fundamental Theorem of Algebra:_______________________________________________________ __________________________________________________________________________________________________ See Below Example 1 on Page 359. Corollary to the Fundament Theorem of Algebra: ________________________ _________________________________________________________________________________________________. We can use this to write equations given zeros of functions.

5. Write the equation of a polynomial function with zeros: 3, -5.

Zeros:__________________

Factored Form: f(x) = __________________

Standard Form: f(x) = _________________________________________________________________

Concept Summary—Zeros, Factors, Roots and Intercepts: See page 358.

Words: Let P(x) be a polynomial function. The following statements are all equivalent.

1. c is a Zero of P(x)

2. c is a __________________ (or solution) of the equation ____________________

3. (_________) is a factor of P(x)

4. (c,0) is an x-intercept of the graph of P(x) if ________________________________

Example: f(x) = x3 – 8x2 – 5x + 84. Use the graph of f(x) to give the following information

Zeros:_____________________________________

The solutions to the equation x3 – 8x2 – 5x + 84 = 0 are:____________________________

What are the factors of f(x)?_________________________________________

What are the x-intercepts of the graph of f(x)?_______________________________________

27

6. Write the equation of a polynomial function with zeros -3, 5, 8. See Page 362: Exercises: Write a polynomial function of least degree with the given zeros. 7. 1 and 5 – i If 5 – i is a zero, then _____________ is also a zero. So the factors are:_______________________________________ Write the polynomial as a product of its factors:_________________________________________________ Multiply to write in standard form: Write an equation of a polynomial function of least degree with the given zeros. 8. 4, 7i 9. -1, 9, -3i 10. 4i, -2i 11. 3, 2 + 3i 12. 0, 1 – i 13. 1, 3, 2 – i

Words: Complex Conjugates Theorem:______________________________________________________ _____________________________________________________________________________________________. Example: If 3 – 2i is a zero, then __________________ is also a zero.

If -5i is a zero, then _______ is also a zero.

28

Graphs of Polynomial Functions

Leading Coefficient

Constant Function

Linear Function Quadratic Function

Cubic Function Quartic

Function

Sketch a graph if a > 0 (leading coefficient is positive).

Write the end behavior of each of the above graphs.

Sketch a graph if a < 0 (leading coefficient is negative).

Write the end behavior of each of the above graphs.

Sketch a graph with the following characteristics. Write an equation in standard form for the graph.

1. Zeros at 4, -1, 3, Degree 5, and with end behavior of as 𝑥 →∞, 𝑓(𝑥) → −∞ and as 𝑥 → −∞, 𝑓(𝑥) →∞.

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

29

Sketch a graph with the following characteristics. Write an equation in standard form for the graph.

2. Zeros at −1

2, 5, 10, and 𝑖, Degree 5, with end behavior as 𝑥 →∞, 𝑓(𝑥) → −∞ and as 𝑥 → −∞, 𝑓(𝑥) →∞.

3. Zeros at -4, -4, 3, 5, Degree 4, with end behavior as 𝑥 →∞, 𝑓(𝑥) → −∞ and as 𝑥 → −∞, 𝑓(𝑥) → −∞.

4. Zeros at -2, 3, and 𝑖, Degree 4, with end behavior as 𝑥 →∞, 𝑓(𝑥) → −∞ and as 𝑥 → −∞, 𝑓(𝑥) → −∞.

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

1 2 3 4 5 6 71 2 3 4 5 6 7

1

2

3

5

6

4

7

1

2

3

5

6

4

7

-6 -4 -2-6 -4 -2

-6

-2

-4

-6

-2

-4

30

5.7 Graphing Polynomial Inequalities Solving Polynomial Inequalities – What is the domain where y is greater than zero? Less than zero? Solving polynomial inequalities involves finding zeros, similar to what we needed to solve polynomial equations. However, our solutions will be in the form of ______________________ instead of x = _____. How to Solve Polynomial Inequalities:

1. Write polynomial in standard form 2. Identify the zeros 3. Determine intervals of interest (above or below zero)

Examples:

1. 𝑥4 − 5𝑥2 − 36 > 0 2. 𝑥3 + 3𝑥2 − 𝑥 < 3 3. 𝑥3 − 2𝑥2 + 𝑥 > 0