Your browser does not have JavaScript enabled. All Apex Learning products require JavaScript. 3.6.6 Test (TS): Polynomial and Rational Functions Test Precalculus Sem 1 (S4201737) Points possible: 100 Nikolaus Lemke Date: ____________ Answer the following questions using what you've learned from this unit. Write your responses in the space provided. 1. Use your calculator to draw the graph of . Part I: Transfer the graph from your calculator to the graph shown below. (4 points) Part II: a. Identify the local maximum by circling the point on the graph. (2 points) b. Identify the local minimum by putting a small square around the point on the graph. (2 points) c. Use your calculator's minimum function to find the coordinates of the absolute minimum of this graph. Round the coordinates to the nearest hundredth, and write your answer in the form (x,y). (2 points) 2. Consider the function . Part I: Factor the numerator of this function. What do you notice about the numerator and denominator of the function? (2 points) Part II: Simplify H(x) and write the new function below as G(x). (2 points) Part III: What significant graphical feature is produced when a function like H(x) can be simplified to G(x) ? (2 points) Part IV: Graph the function H(x), and clearly state its domain and its range, taking note of the fact that H(x) is not the same as G(x). (4 points) 3. Divide the following polynomials using the long division model: . Part I: Express this problem using the standard format for a problem of (2 points) Part II: Use this checklist to proceed through this problem: (8 points) How many times does x2 go into the largest term in the problem? Write this value on top of the problem and multiply that value by x2 + x + 1 Write that product below the lowest line on your work and subtract it from what remains in the problem. Continue this process until you can no longer divide x2 into what remains in the problem. Include your remainder in the final answer. 4. Divide, using synthetic division: . Part I: List the coefficients of the dividend in order. (2 points) Part II: What value should you use to represent the divisor? (2 points) Part III: Carry out the division problem, and list the coefficients of the quotient in order. (4 points) Part IV: Convert the coefficients you found into the final answer to the question. (2 points) 5. For the function: Part I: a. What value of x makes this function undefined? (2 points) b. Write the equation of the vertical asymptote defined by this value of x. (2 points) Part II: a. Which expression has a larger degree, the numerator or denominator? (2 points) b. Use this information and your choice of long or synthetic division to determine the equation of the oblique asymptote for this function. (4 points) 6. Use Descartes' rule of signs to describe the roots of . Part I: How many total roots must there be in this fourth-degree function? (2 points) Part II: a. How many sign changes are there in the original function H(x)? (1 point) b. How many positive real roots are possible? (2 points) Part III: a. Replace every x with (-x) and rewrite the function below. (2 points) b. How many sign changes are there in this new function? (1 point) c. How many negative real roots are possible? (2 points) Part IV: In light of the number of total roots for this function, how many possible complex roots are there? (2 points) For problems 7 - 8, solve each of the polynomial equations. 7. Part I: Make a list of possible rational roots. (2 points) Part II: Test the possible roots until you find one that produces a remainder of 0. (4 points) Part III: Write the resulting quadratic function. (2 points) Part IV: Factor or use the quadratic formula to find the final 2 roots. (2 points) 8. Part I: Make a list of possible rational roots. (2 points) Part II: Test the possible roots until you find one that produces a remainder of 0. (2 points) Part III: Write the resulting cubic function. (2 points) Part IV: Use synthetic division to find a second root that will reduce the cubic expression to a quadratic expression. (2 points) Part V: Use the quadratic formula to find the final 2 roots. (2 points) 9. Write a polynomial with rational coefficients having roots 3, 3 + i, and 3 - i. Part I: Write the factors (in the form x - a) that are associated with the roots (a) given in the problem. (2 points) Part II: Multiply the 2 factors with complex terms to produce a quadratic expression. (4 points) Part III: Multiply the quadratic expression you just found by the 1 remaining factor to find the resulting cubic polynomial. (4 points) 10. List the possible rational roots for . Part I: a. What is the value of the constant term? (1 point) b. List all of the constant term's factors. (2 points) Part II: a. What is the value of the leading coefficient? (1 point) b. List all of the leading coefficient's factors. (2 points) Part III: Use your answers to Part I b and Part II b to find the possible rational roots. (2 points) Copyright 2014 Apex Learning Inc. Use of this material is subject to Apex Learning's Terms of Use . Any unauthorized copying, reuse, or redistribution is prohibited. Apex Learning and the Apex Learning Logo are registered trademarks of Apex Learning Inc