PreCalculus 4-R Unit 4 Polynomial and Rational Functions Review Problems

Sketch the graph of the function. No Graphing Calculator 1

Review Problems Sketch the graph of the function. No Graphing Calculator 2

Review Problems Sketch the graph of the function. No Graphing Calculator 3

Review Problems Sketch the graph of the function. No Graphing Calculator 4

Review Problems Sketch the graph of the function. No Graphing Calculator 5

Review Problems No Graphing Calculator Find the quotient and remainder using division. The quotient is (x + 1); the remainder is –38. The quotient is the remainder is 5 6

Review Problems Find the quotient and remainder using division. The quotient is ; the remainder is 5 No Graphing Calculator The quotient is 1; the remainder is (2x + 3). 7

Review Problems Evaluate P(3) for: 38 No Graphing Calculator Evaluate P(2) for: 234 Evaluate for: 8

Review Problems Find a polynomial of degree 5 and zeros of –6, –2, 0, 2, and 6 No Graphing Calculator 9

Review Problems Find a polynomial of degree 3 that has zeros of 2, –4, and 4, and where the coefficient of x 2 is 6. No Graphing Calculator 10

Review Problems Find the polynomial of degree 4 whose graph is shown. 11

Review Problems List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). No Graphing Calculator 12

Review Problems Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. 0, 2 or 4 negative No Graphing Calculator 13

Review Problems A polynomial P is given. Find all the real zeros of P. Sketch the graph of P. No Graphing Calculator 14

Review Problems Find all rational zeros of the polynomial. No Graphing Calculator Find all rational zeros of the polynomial. 15

Review Problems Find all rational zeros of the polynomial. No Graphing Calculator Find all rational zeros of the polynomial. 16

Review Problems Find all of the real zeros of the polynomial and sketch the graph of P No Graphing Calculator 17

Review Problems Evaluate the expression (9 + 14i) + (7 – 11i) and write the result in the form a + bi i Evaluate the expression 10(–2 + 14i) and write the result in the form a + bi. – i 18

Review Problems Evaluate the expression and write the result in the form a + bi i Evaluate the expression and write the result in the form a + bi. 2 – 15i 19

Review Problems Evaluate the expression i 17 and write the result in the form a + bi. i Evaluate the expression and write the result in the form a + bi. 2i2i 20

Review Problems Evaluate the expression and write the result in the form a + bi. –18 Find all solutions of the equation x 2 – 8 x + 25 = 0 and express them in the form a + bi. x = 4 + 3i, x = 4 – 3i 21

Review Problems Find all solutions of the equation and express them in the form a + bi. z = –4 + 2i, z = –4 – 2i A polynomial P is given. Factor P completely. 22

Review Problems Find the polynomial P(x) of degree 3 with integer coefficients, and zeros 4 and 3i. Factor the polynomial completely and find all its zeros. 23

Review Problems Factor the polynomial completely into linear factors with complex coefficients Find the x- and y-intercepts of the rational function x-intercept (6, 0), y-intercept (0, –1) 24

Review Problems Find the x- and y-intercepts of the rational function x-intercept (–3, 0), y-intercept (0, 3) Find the horizontal and vertical asymptotes of the rational function horizontal asymptote y = 0; vertical asymptote x = –8 25

Review Problems Determine the correct graph of the rational function No Graphing Calculator 26

Review Problems Determine the correct graph of the rational function No Graphing Calculator 27

Review Problems Determine the correct graph of the rational function No Graphing Calculator 28

Review Problems Use a Graphing Calculator to find vertical asymptotes, x- and y- intercepts, and local extrema, correct to the nearest decimal. (a) Find all vertical asymptotes. (b) Find x-intercept(s). (c) Find y-intercept(s). (d) Find the local minimum. (e) Find the local maximum. x = 2; x = 0 y = 0 (2.7, 12.3) none 29

Review Problems Find the intercepts and asymptotes (a) Determine the x-intercept(s). (b) Determine the y-intercept(s). (c) Determine the vertical asymptote(s). (d) Determine the horizontal asymptote(s). no solution y = –3 x = –1, x = 3 y = 3 30

Review Problems Find all horizontal and vertical asymptotes (if any). (a) Find all horizontal asymptotes (if any). (b) Find all vertical asymptotes (if any). no solution x = 2, x = –2 31

Review Problems Find the intercepts and asymptotes (a) Determine the x-intercept(s). (b) Determine the y-intercept(s). (c) Determine the vertical asymptote(s). (d) Determine the horizontal asymptote(s). x = 2 y = –1 x = –8 y = 4 32

Review Problems Graph the polynomial, find the x and y intercepts, show the end behavior 33

Review Problems Graph the polynomial, find the x and y intercepts, show the end behavior x intercepts y intercept 1 34

Review Problems Graph the polynomial, find the x and y intercepts, show the end behavior x intercepts y intercept 35

Review Problems Graph the polynomial, find the x and y intercepts, show the end behavior x intercepts y intercept 3 36

Answers The quotient is (x + 1); the remainder is –38. The quotient is the remainder is 5 The quotient is ; the remainder is 5 The quotient is 1; the remainder is (2x + 3) , 2 or 4 negative

Answers A 27. C 28. C i – i i2 – 15i i 2i2i –18 x = 4 + 3i, x = 4 – 3i z = –4 + 2i, z = –4 – 2i x-intercept (6, 0), y-intercept (0, –1) x-intercept (–3, 0), y-intercept (0, 3) horizontal asymptote y = 0; vertical asymptote x = –8 x = 2; x = 0 y = 0 (2.7, 12.3) none no solution y = –3 x = –1, x = 3 y = 3 no solutionx = 2, x = –2

Answers x = 2 y = –1 x = –8 y = 4 x intercepts y intercept 1 x intercepts y intercept x intercepts y intercept 3