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HOMEWORK PACKET ~ UNIT 5 “POLYNOMIAL FUNCTIONS” Name________________________ Date ____________ Day 1 Homework Factoring Polynomials by GCF, Sum/Difference of Cubes, Operations with Polynomials I. REVIEW – Factor all of these completely 1. 16x 2 y 2 2. 3. 8 x 2 50 y 2 4. 8 x 2 50 y 2 3x3 10 x 2 8 x II. FACTOR 5. x3 – 27 6. x3 + 27 7. x3 – 64 8. 2x3 + 16 9. 8 x3 27 11. 64 x 1 3 10. 8 x3 27 12. 8m 64n 3 3 III. Complete the table Polynomial: YES or NO Function Standard form of the polynomial Degree of polynomial Leading Coefficient of the polynomial 13. y = -3x2 + 8x4 - 2 5x 1 14. y x3 x 2 7 15. f ( x) 3x 8 x 5 x 5 8 16. g(x) = - 7x2 + 7x -3 IV. Add, subtract, multiply or divide to simplify. Write your answer in standard form. 17. (7q 3q3) + (16 8q3 + 5q2 q) 18. (4z4 + 6z 9) + (11 z3 + 3z2 + z4) 19. (l0v4 2v2 + 6v3 7) (9 v + 2v4) 20. (4x5 + 3x4 5x + l) (x3 + 2x4 x5 + 1) 21. 2x3(5x 1) 22. (n + 5)(2n2 n 7) 24. (2x + 5)3 23. (x - 3)3 Constant Name________________________ Date ____________ Day 2 Homework: Function Composition 1. f(x) = 3x – 5 Find f o g 4. f(x) = 2x2 + 11 g(x) = 2x + 1 [this means f(g(x)] g(x) = 4x Find g(f(x)). 7. f(x) = -4x + 6 Find f o g (2) 2. f(x) = x – 2 g(x) = x2 + 3 Find g o f 5. f(x) = x2 – 9 [this means f(g(2)] 8. f(x) = 2x2 g(x) = 3x Find f(g(x)) g(x) = 3x2 Find f o g (x) g(x) = 5x – 1 3. f(x) = 5x – 6 6. f(x) = 5x2 Find f o f (x) g(x) = x2 + 7 Find f(g(-1)) and find g(f(-2)) 9. f(x) = -3x + 5 Find f o g g(x) = 6x – 2 and find g o g (x) g(x) = -2x 10. f(x) = x – 3 g(x) = x2 Find g o f 13. f(x) = x2 + 10 g(x) = -4x Find f(g(x)). g(x) = 2x2 Find f(g(x)) 16. f(x) = x2 + 3x 11. f(x) = 6x – 5 g(x) = x2 + 6 g(x) = 4x – 3 and find g o g 17. f(x) = 2x 4 Find f(g(-2)). Also find g(f(-2)). Find f o g g(x) = 7x + 1 Find g(f(x)). 14. f(x) = 3x2 Find f o f 12. f(x) = x2 – 9 g(x) = 1 x2 2 and find g o f 15. f(x) = 4x – 6 g(x) = 3x – 2 Find f(g(3)) 18. k(x) = 2 x 6 Find k o h h(x) = x 4 and find h o k Name________________________ Date _____________ Day 3 Homework: Graphing Cubic Functions I. REVIEW #1 & 2 Simplify each of the following. 1. 1 4 2 x 3x 1 3 2. 6 4 3 1 2 6 (x y z ) #3-5 Solve each of the following by the zero product property (factor). 3. 3x2 – 27 = 0 4. 4x2 – 9 = 0 5. 2x2 – 2x – 12 = 0 II. PRACTICE #6-9 Graph each of the following. Identify the Point of Inflection and describe the transformations. 6. f x x 1 7. y 3 y 1 3 x 1 2 Point of inflection_________ y Point of inflection_______ Horizontal shift: ___________ Horizontal shift:_________ x Vertical shift: _____________ Reflection: _______________ x Stretch/shrink: ____________ Reflection: ____________ Stretch/shrink: _________ 8. y x 1 2 9. g x 2 x 3 1 3 3 y y Point of inflection_________ Point of inflection_______ Horizontal shift: ___________ x x Vertical shift: __________ Vertical shift: _____________ Reflection: _______________ Stretch/shrink: ____________ Horizontal shift:_________ Vertical shift: __________ Reflection: ____________ Stretch/shrink: _________ Name________________________ Date ________ Day 4 Homework: End Behavior, Zeros and Multiplicity State the degree of each polynomial and the leading coefficient. Describe the end behavior of the graph of the polynomial function. 1. f(x) = 2x5 7x2 4x Degree: Leading coefficient: As x f(x) ______ and as x + f(x) ______ As x f(x) ______ and as x + f(x) ______ As x f(x) ______ and as x + f(x) ______ As x f(x) ______ and as x + f(x) ______ 2. f(x) = 9x7 + 2x8 + 10 Degree: Leading coefficient: 3. f(x) = - 95x50 + 407x2013x80 Degree: Leading coefficient: 4. f(x) = 2017x57 1998x46 + 1999x23 Degree: Leading coefficient: Describe the degree (even or odd) and leading coefficient (positive or negative) of the polynomial function. Then describe the end behavior of the graph of the polynomial function. 5. 6. Degree _______ Degree _______ Leading Coefficient __________ Leading Coefficient __________ As x then f ( x) _____ As x then f ( x) _____ As x then f ( x) _____ As x then f ( x) _____ #7-9 Use your calculator to match the graph with its function. 7. f(x) = 2x 4 3x 2 2 A. 8. f(x) = 2x 6 6x 4 + 4x 2 2 B. 9. f(x) = 2x 4 + 3x 2 2 C. #10 & 11 Given the graph, complete the following. 10. f ( x) = x3 - 4 x 2 - 3x + 18 a. Degree __________ b. Lead coefficient: positive or negative c. Zeros ______________ Multiplicity? _____________ d. Factors __________________________ e. Number of turning points _____________________ f. Identify all relative minimum/maximum points. _______________________________________ g. Identify all absolute minimum/maximum points. _______________________________________ h. Increasing intervals _________________ i. Decreasing intervals _________________ j. y-intercept ________________________ 11. g ( x) = x 3 - 7 x - 6 a. Degree __________ b. Lead coefficient: positive or negative c. Zeros ______________ Multiplicity? _____________ d. Factors __________________________ e. Number of turning points _____________________ f. Identify all relative minimum/maximum points. _______________________________________ g. Identify all absolute minimum/maximum points. _______________________________________ h. Increasing intervals _________________ i. Decreasing intervals _________________ j. y-intercept ________________________ Name________________________ Date ________ Day 5 Homework: Evaluating Polynomials with Synthetic Substitution and The Fundamental Theorem of Algebra I. REVIEW – Simplify 1. 1 4 2 x 3x 1 3 2. 6 4 3 1 2 6 (x y z ) Solve by the Zero Product Property (factor) 3. 3x2 – 27 = 0 4. 4x2 – 9 = 0 5. 2x2 – 2x – 12 = 0 II. Evaluate the polynomial for the given value of x in TWO ways – by direct substitution and by synthetic substitution 6. f(x) = 5x3 – 2x2 – 8x + 16 for x = 3 Direct Substitution Synthetic Substitution 7. f(x) = 8x4 + 12x3 + 6x2 – 5x + 9 for x = -2 Direct Substitution Synthetic Substitution 8. f(x) = x3 + 8x2 – 7x + 35 for x = –6 Direct Substitution Synthetic Substitution 9. f(x) = –8x3 + 14x – 35 for x = 4 Direct Substitution Synthetic Substitution 10. f(x) = –2x4 + 3x3 – 8x + 13 for x = 2 Direct Substitution Synthetic Substitution #11-15 Identify the total number of zeros and maximum number of turning points. 11. f(x) = 5x3 – 2x2 – 8x + 16 ZEROS:_________ Max. # of turning points: ___________ 12. f(x) = 8x4 + 12x3 + 6x2 – 5x + 9 ZEROS:_________ Max. # of turning points: ___________ 13. f(x) = x7 + 8x4 – 7x + 35 ZEROS:_________ Max. # of turning points: ___________ 14. f(x) = –8x3 + 14x – 35 ZEROS:_________ Max. # of turning points: ___________ 15. f(x) = –2x6 + 3x3 – 8x + 13 ZEROS:_________ Max. # of turning points: ___________ III. Write a polynomial function in standard form of least degree that has a leading coefficient of 1 and the given zeros. (Remember, imaginary and irrational solutions always come in pairs! You may have to find the other half of the pair!) 16. -2, 1, 3 17. -5, -1, 2 18. 2, -i, i 19. 2, -3i 20. 4, 5 , 5 21. 3, 1 2 22. Graph f(x) = 3x4 + x3 - 10x2 + 2x + 7 using your calculator. Sketch its graph below. Determine the total number of zeros for the polynomial __________ # of real zeros:__________ # of imaginary zeros:____________ a. Determine the number of turning points _________________________ b. Identify all relative minimum/maximum points. _____________________________________________ c. Identify all absolute minimum/maximum points. _____________________________________________ d. Over what intervals is f(x) Decreasing___________________________________________ e. Over what intervals is f(x) Increasing___________________________________________ f. Describe the end behavior of the graph: As, x f ( x) _____ As x , f ( x) _____ Name________________________ Day 6 Homework: Applying the Remainder and Factor Theorems I. REVIEW – Multiply 1. ( x 5)( x 5) Date ________ 2. ( x 2i )( x 2i ) II. PRACTICE – Use SYNTHETIC DIVISION and LONG DIVISION to divide the polynomials. Be sure to write your answer in the form of a polynomial and a remainder. SYNTHETIC DIVISION 3. (x3 3x2 + 8x 5) (x 1) 4. (x4 7x2 + 9x 10) (x 2) 5. (2x4 x3 + 4) (x + 1) 6. (2x4 11x3 + 15x2 + 6x 18) (x 3) LONG DIVISION III. Factor the following polynomials completely using synthetic division and factoring. 7. f(x) = x3 3x2 16x 12; given that (x 6) is a factor 8. f(x) = x3 12x2 + 12x + 80; given that (x – 10) is a factor 9. f(x) = x3 18x2 + 95x 126; given that (x – 9) is a factor 10. f(x) = x3 x2 21x + 45; given that (x + 5) is a factor 11. f(x) = x3 + 2x2 - 20x + 24; given that 6 is a zero 12. f(x) = 15x3 119x2 l0x + 16; given that 8 is a zero 13. f(x) = 2x3 + 3x2 39x 20; given that 4 is a zero Name________________________ Date ___________ Day 7 Homework: Finding All Rational Zeros #1-4 Use the function, g(x) = x3 – 5x + 4 to answer the questions. 1. List all possible rational zeros of g(x): 2. Graph g(x) on your calculator. Sketch the graph. Pick a zero that matches a value from the list above. Test that zero using synthetic division: 3. Solve the depressed polynomial, using your method of choice (factoring, quadratic formula, or completing the square). 4. List all of the zeros of g(x): #5-9 Use the function, h(x) = 2x3 + 2x2 – 8x – 8 to answer the questions. 5. List all possible rational zeros of h(x): 6. Graph h(x) on your calculator. Sketch the graph. Pick a zero that matches a value from the list above. Test that zero using synthetic division: 7. Solve the depressed polynomial, using your method of choice (factoring, quadratic formula, or completing the square). 8. List all of the zeros of h(x): 9. Solve 0 = 2x3 + 2x2 – 8x – 8 by factoring. (Hint: How do you factor a polynomial with 4 terms?) Does this answer match your answer from #8? 10. a) Is it possible for a cubic function to have more than three real zeros? Explain. (Your explanation can include a picture). b) Is it possible for a cubic function to have no real zeros? Explain. (Your explanation can include a picture). 11. How many possible solutions (real or imaginary) are guaranteed by the Fundamental Theorem of Algebra for the following equation? y = x5 – 3x4 – 5x3 + 15x2 + 4x – 12 Maximum number of turning points? 12. List all possible rational zeros for the function in question 11: 13. Find all the solutions for the function in question 11. (Hint: They are ALL real! You will have to do synthetic division 3 times using zeros from your graph before you have a quadratic to solve.) #14-18: Use your calculator and synthetic division to find all possible solutions. Remember, complex numbers are also solutions. 14. y = x3 -6x2 +11x -6 15. f(x) = x4 -7x2 +12 16. f (x) = x3 -9x2 +20x -12 17. y = x5 -7x4 +10x3 + 44x2 -24x Name________________________ Date ________ Day 8 Homework: REVIEW Analyzing Graphs Answer the questions based on the given graph. Leading Coefficient: (positive or negative) ___________________ End Behavior: As x , f ( x) ____ 9 y 8 7 6 5 4 3 2 1 -1 -9-8 -7 -6 -5-4 -3 -2 -1 -2 -3 -4 -5 -6 -7 -8 -9 1 2 3 4 5 6 7 8 9 As x , f ( x) ____ Identify the real zeros of the graph: ___________________________________________ x Identify the factors based on your zeros listed above. ___________________________________ Circle the turning points on the graph. Determine if they are relative maximums or minimums, absolute maximums or minimums. Determine the intervals where the polynomials are Increasing: ______________________________ Decreasing: _____________________________ Determine the domain and range of the polynomial. Domain: _________ Range: _______________ Leading Coefficient: (positive or negative) ___________________ End Behavior: As x , f ( x) ____ As x , f ( x) ____ Identify the real zeros of the graph: 9 y 8 7 6 5 4 3 2 1 -1 -9-8 -7 -6 -5-4 -3 -2 -1 -2 -3 -4 -5 -6 -7 -8 -9 1 2 3 4 5 6 7 8 9 ___________________________________________ Identify the factors based on your zeros listed above. ___________________________________ x Circle the turning points on the graph. Determine if they are relative maximums or minimums, absolute maximums or minimums. Determine the intervals where the polynomials are Increasing: ______________________________ Decreasing: _____________________________ Determine the domain and range of the polynomial. Domain: _____________ Range: _______________ Operations & Substitution Simplify each expression. Show work! 1. (x + 1)3 2. (2x4 − 8x2 − x) − (−5x4 − x + 5) 3. (5d3 – 4d2 + 5) + (7d3 + 2d2 – 8d) 4. (x + 5i)(x – 5i) 5. (2x + 3)(x – 2)(3x + 2) 6. (x + 4)(x2 + 2x – 3) 7. Evaluate the polynomial function using Direct Substitution. f(x) = -3x3 + x2 – 12x – 5 when x = -2 8. Evaluate the polynomial function using Synthetic Substitution. F(X) = x4 + 2x3 + 5x - 8 for f(-4) 9. Write a polynomial function in standard form that has real coefficients, the given zeros, and a leading coefficient of 1. Zeros: 2, 4, -3i Recognizing and "Reading" Polynomials Identify the degree, leading coefficient, and constant of the polynomial. (State the numerical value.) 10. f(x) = 6x5 – 4x3 + 1 Degree _____ Leading Coefficient _____ Constant ____ 11. g(x) = 9x4 + x – 7 Degree _____ Leading Coefficient _____ Constant ____ 12. f(x) = -2x2 – 3x4 + 5x – 9x3 + 5 Degree _____ Leading Coefficient _____ Constant ____ Tell if each of the following are polynomials? If no, explain why not! (Yes or No) Explanation 13. 4x2 + 2x + x2 + 3 _________ _____________________ 14. 3x + 1 _________ _____________________ 15. 5x + 2x½ +3 _________ _____________________ Describe the end behavior of the graph. Use or 16. f(x) = -7x2 + 4x – 9 as x , f ( x) _____ as x , f ( x) _____ 17. f(x) = 6x5 + 7x4 – 8x as x , f ( x) _____ as x , f ( x) _____ 18. f(x) = -8x3 – 9x2 + x – 4 as x , f ( x) _____ as x , f ( x) _____ as x , f ( x) _____ as x , f ( x) _____ 19. f(x) = x4 + 16 Factoring Factor the following completely. If not possible, write PRIME. 20) 2x3 + 16 21) 25x2 – 81 22) 125x3 - 27 23) 2x4 + 128x Composition of Functions For Problems 24-29, let: f(x) = 2x-2, g(x) = 3x, h(x) = x2 +1 24. f(g(-3)) 25. f(h(7)) 26. (g(f(x)) 27. f(x+1) 28. h(g(x)) 29. (f ᵒg)(x) Synthetic Substitution Evaluate the function at the given value, then determine if the value given is a solution. 30. f(x) = x4 + 2x3 – 13x2 + 15x + 22, x = -5 31. f(x) = x5 + x4 – 15x3 – 19x2 – 6x + 1, x = 4 32. g(x) = -5x5 + 11x4 + 9x3 + 11x2 – 8x + 4, x = 3 Finding Solutions (33-34) Use your calculator and synthetic division to find all solutions of the given equations. Remember, complex numbers are also solutions. 33. y = x4 - 2x3 – 19x2 + 32x + 48 34. y = x5 + 4x4 – 10x3 + 82x2 – 375x - 450 Use long division to simplify the polynomial, then find all of the zeros of f(x). 35. f(x) = (12x3 + 2 + 11x + 20x2) ÷ (2x + 1)