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Pre-Calculus Unit 3: Polynomial FunctionsVocabulary:Objectives:Polynomial Function:Degree:End behavior:+ LC- LCEven degreeOdd degreeZeroes:Multiplicity:Even multiplicityOdd multiplicityReal vs. imaginary:Rational zeroes theorem:Conjugate zeroes theorem:Remainder Theorem:Long division:Synthetic division:Analyze a polynomial function when given a graph or equation (in particular the zeroes, end behavior, and degree).Create polynomial functions from a variety of given information (graphs, properties).Apply polynomial functions to real life situations.What do they look like???y=#y=mx+by=ax2+bx+cy=ax3+bx2+cx+dy=ax4+bx3+cx2+dx+ey=ax5+bx4+cx3+dx2+ex+fy=ax6+bx5+cx4+dx3+ex2+fx+gy=ax7+bx6+cx5+dx4+ex3+fx2+gx+hThe pattern continues…Properties of Polynomial FunctionsUse the zeroes and any additional information to graph each polynomial function.1. fx=-(x-1)2(x-2)2. fx=-2x4-x3+3x2Use the zeroes and any additional information to write an equation for each polynomial function.3. Zeroes at x = 3 (M2), x = -2 (M1), and x = 5 (M1); end behavior similar to fx=-x2.4. THINK ABOUT: Is it important to be able to factor a polynomial? Why or why not?Is it better to write a polynomial in standard form or factored form? Support.More About Zeroes…Find the zeroes of the function, fx=x5+x4+x3+x2-20x-20.Any Luck?GraphingLong DivisionSynthetic DivisionConclusion: _____________________________________________________________________Determine whether x-3 is a factor of fx=2x3-7x2+5. Support in more than one way.Finding ZeroesFind all of the zeroes of the function, fx=x3-3x+2 as if it were…Year: 1915Year: 2018Zeroes:Factored polynomial:Zeroes:Factored polynomial:Find the zeros of each polynomial using any appropriate strategy.1. Px=x4-5x3-5x2+23x+102. Px=2x3-7x2+4x+4Even More About (COMPLEX) Zeros 1. Graph the function, fx=x3-3x2+x-3 and explain why the graph is not sufficient for finding all of the zeros. How can we find the remaining zeros?Zeros:Factored Polynomial:2. GIRLS: fx=x3-2x+4A) All zeros and their multiplicities:B) Factored polynomial:3. GUYS: fx=3x5+24x3+48xA) All zeros and their multiplicities:B) Factored polynomial:THINK ABOUT: What do you notice about the complex solutions?Polynomial InequalitiesSteps: Get the whole equation on one side.Find the zeros.Figure out the sign of the polynomial on either side of the zeros.Write solution.Check on a graph!x2-10<3xx2-5x≥-6x3>-x2+12xx4+4x3-12x2≤0Applications of Polynomial Functions1. WCS students were excited when snow began to fall on Sunday at noon. The meteorologist said the amount of snow (in inches) could be modeled by the function, At=11.6t-12.41t2+6.2t3-1.58t4+0.2t5-0.01t6 where t is the time in days. A) If crews are under contract to removed snow from the parking lots when/if the accumulation reaches 2 inches, when will they need to begin snow removal?B) What happened shortly after noon on Tuesday?C) You’re hoping for at least 5 inches of snow (for great sledding). Does this happen? If so, when?D) On what day (and to the nearest hour) did all of the snow melt?2. The weight (measured in carats) of a round cut diamond can be modeled by the function, Cx=0.006x3-0.081x2+0.58x, where x is the diameter of the diamond in millimeters.A) What is the weight of a diamond with diameter 3 mm?B) If a jeweler wants to set a 5 carat ring, what size diamond does he need to purchase? UNIT 3 PRACTICE1. Let fx=2x-1x+3x-5A) Find all real zeros and state the multiplicity of each:B) Find the y-intercept:C) State the end behavior:D) Estimate the local extrema:2. Let fx=x-22x+3A) Find all real zeros and state the multiplicity of each:B) Find the y-intercept:C) State the end behavior:D) Range:E) fx=-33. Let fx=6x3+3x2+1A) Increasing intervals:B) Local extrema:C) Is (x+1) a factor? Support.D) f-1=4. Let fx=x3-3x2+3x-1A) Is (x-1) a factor? Support.B) Find f(2) without plugging 2 into the function.C) Find all zeros.D) Factored polynomial: 5. Let fx=x5+x3+8x2+8A) Find all zeros.B) Factored polynomial:*hint—try to factor by grouping6. Let f(x) be represented by the graph below.A) Zeros:B) Potential equation:C) Graph part B in your calculator. How is similar to the one at left? Different? Why? How could you get the exact equation for the graph at left?7. Let f(x) be a function with real zeros: 2, -4, and 5; all with multiplicity 1.A) Is there more than one way to sketch the function? Explain.B) Can you accurately predict the end behavior?C) Does this have to be a third degree function? Explain. D) Write a potential equation in factored form.8. According to USPS a parcel can measure no more than 108 inches in length and girth combined. The length is designated to be the longest side and girth is the distance around a cross section perpendicular to the length. A) Assuming the cross section is a square, sketch the parcel.B) Write an equation to model the volume.C) What is the maximum volume?D) What are the dimensions of the parcel that produces the max volume?9. Given the function, fx=x3-x2-14x+11A) Graph the function and determine a situation that could be modeled from the function. B) Write and answer 3 different types of questions about the model. Questions should be written in the context of the problem.10. Solve the following inequalities:x2+5x-14≥0x4-4x3<12x211. Write an equation for a polynomial function (in factored form) that:Has end behavior similar to fx=-x4Has two complex zeros12. Write an equation for a polynomial function (in standard form) that:Is third degreeCan be completely solved by factoringProvide a short response to each question.13. “Everything in this unit can be done on a calculator.” Do you agree or disagree? Support.14. Describe how you would go about finding the zeros of any polynomial. Be sure your description accounts for all types of zeros.15. Do you need to know how to do both long and synthetic division? Do both strategies work for all types of polynomial division? ................
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