Math 121. Practice Problems from Chapter 3 Fall 2016

Transcription

1 Math 121. Practice Problems from Chapter 3 Fall 2016 Section 1. Remainder and Factor Theorems 1. Divide polynomials using long division. For practice see Exercises 1, Synthetic division. For practice see Exercises 3, Use the remainder theorem and factor theorems. For practice see Exercises 5, Use synthetic division to write a polynomial as the product of the divisor and quotient. For practice see Exercise 7 Section 2. Polynomials of Higher Degree 1. Determine the far right and far left behavior of a polynomial. For practice see Exercises 1, 2 2. Use the intermediate value theorem to determine if a polynomial has a zero between two values. For practice see Exercise Find the real zeros of a (simple) polynomial by factoring. For practice see Exercise Find the x-intercepts of a polynomial and determine whether the graph of a polynomial crosses the x-axis at a given intercept based on the power of the factor. For practice see Exercise Find intercepts and sketch a rough graph of a factored polynomial. For practice see Exercise 6. Section 3. Zeros of Polynomial Functions 1. Find zeros with their multiplicities of a polynomial. For practice see Exercise Identify zeros and properties of a polynomial from its graph. For practice see Exercise Use the rational zero theorem to determine possible rational zeros of a polynomial. For practice see Exercise Use Decartes rule of signs theorem to determine possible numbers of positive and negative real zeros of a polynomial. For practice see Exercise Find all zeros of a given polynomial using synthetic division. For practice see Exercises 5, Find a polynomial of smallest degree with a given set of zeros. For practice see Exercises 7, 8.

2 Section 4. The Fundamental Theorem of Algebra 1. Find all zeros real and complex of a given polynomial. For practice see Exercises 1, Find a polynomial of smallest degree with a given set of zeros. For practice see Exercises 3. Section 5. Rational Functions 1. Determine the domain of a rational function. For practice see Exercise Find the domain, x-intercepts, y-intecept, and vertical asymptotes of rational function. For practice see Exercise Find the vertical asymptotes of a rational function, and describe the behavior of the rational function just to the right and just to the left of the asymptote. For practice see Exercise Determine difference between a rational function having a vertical asymptote and having a hole in its graph. For practice see Exercise 4, 5, Find horizontal or slant asymptotes of a rational function. For practice see Exercise 7, Determine domain of rational function, find all asymptotes (horizontal, vertical, slant), find x- and y-intercepts, plot additional points and then graph a rational function. For practice see Exercise 9, 10. Page 2

3 1 Remainder and Factor Theorem 1. Use long division to find (2x 4 4x 3 + 2x + 5) (x 2 + 4). 2. Use long division to find (6x 3 2x 2 3x + 2) (2x 2 + 1). 3. Use synthetic division to find (x 4 4x 2 6x + 1) (x 3) 4. Use synthetic division to find (2x 5 + 9x 4 5x 2 5x + 1) (x + 2) 5. Consider the polynomials P and Q defined as follows P (x) = x x 4101 and Q(x) = x x 4082 (a) Use a calculator to find P ( 2), P (2), Q( 2) and Q(2). (b) Use your answers from (a) and the remainder and factor theorems to answer the following questions. What is the remainder of P (x) (x 2)? Is (x 2) a factor of P (x)? What is the remainder of P (x) (x + 2)? Is (x + 2) a factor of P (x)? What is the remainder of Q(x) (x 2)? Is (x 2) a factor of Q(x)? What is the remainder of Q(x) (x + 2)? Is (x + 2) a factor of Q(x)? 6. Ken was trying to factor a polynomial, so he programmed the formula for P (x) in his calculator and he correctly found that P ( 8) = 20, P ( 7) = 0, P ( 6) = 0, P (0) = 14, P (3) = 0, and P (8) = 15. (a) Even with this information Ken was still puzzled, so he asked his brilliant girlfriend JoAnn, and she gave him a hint by telling him that there are three obvious factors from the given information. Is JoAnn correct? If so, what are the factors? (b) Based on the values of P given above. Answer the following questions. (i) What is the remainder of P (x) (x + 8)? (ii) What is the remainder of P (x) (x + 7)? (iii) What is the remainder of P (x) x? (iv) What is the remainder of P (x) (x 8)? 7. Use synthetic division to show that (x + 7) is a factor of the polynomial P given by P (x) = x x x + 7 Then write P as the product of (x + 7) and a quadratic factor. Page 3

4 2 Polynomials of Higher Degree 1. Determine the far right and far left behavior of the polynomial P (x) = 2x x 6 6x 3 + 7x 6. (i) What is the leading coefficient of P? (ii) What is the degree of P? Is it even or odd? (iii) Because the degree of P is coefficient of P is of P is (choose best response) (even or odd) and because the leading (positive or negative) we know the long-term behavior (a) Up to far left and up to far right (b) Up to far left and down to far right (c) Down to far left and down to far right (d) Down to far left and up to far right (e) None of the above. 2. Determine the far right and far left behavior of the four polynomials given below P (x) = 5x x 10 4x 3 + 9x 6. Q(x) = 5x x 10 4x 3 + 9x 6. R(x) = 5x x 10 4x 3 + 9x 6. S(x) = 5x x 10 4x 3 + 9x 6. For each of the polynomials above, state whether its leading coefficient is negative or positive, and state whether its degree is even or odd, and then choose one of the following options. (a) Up to far left and up to far right (b) Up to far left and down to far right (c) Down to far left and down to far right (d) Down to far left and up to far right (e) None of the above. 3. Use the Intermediate Value Theorem to determine whether P has a zero between a and b. P (x) = 3x 3 + 2x 2 3x 4; a = 1, b = 4 First, find P (1) = and P (4) =. Then choose the best response from the following: (a) Because P (1) and P (4) have opposite signs, we know that P has at least one real zero between 1 and 4. (b) Because P (1) and P (4) have opposite signs, we do not know if P has at least one real zero between 1 and 4. Page 4

5 (c) Because P (1) and P (4) have the same sign, we know that P has at least one real zero between 1 and 4. (d) Because P (1) and P (4) have the same sign, we do not know if P has at least one real zero between 1 and Find the real zeros of the polynomial function by factoring; for this polynomial try factoring by grouping. P (x) = x 3 3x 2 16x Determine the x-intercepts for the graph of P. For each x-intercept, use the Even and Odd Powers of (x c) Theorem to determine whether the graph of P crosses the x-axis or intersects but does not cross the x-axis at that intercept. P (x) = (x 3) 2 (x + 6) 5 (x 7) 7 (x + 3) Use the polynomial P given below in both standard and factored form to answer the following questions. P (x) = x 3 5x 2 + 3x + 9 = (x 3) 2 (x + 1) (a) Determine the far right and far left behavior of P. (b) List the x-intercepts, and at each intercept determine whether the graph of P crosses or intersects but does not cross the x-axis. (c) Find the y-intercepts. (d) Use the above information to sketch a rough graph of P. Page 5

6 3 Zeros of Polynomial Functions 1. Find the zeros of the polynomial function, and state the multiplicity of each zero. P (x) = x 2 (2x 3) 4 (x 2 1) 2 2. Use the graph of the polynomial P given below to answer the following questions y x (a) Use the far right and far left behavior to determine whether the degree of P is even or odd. (b) Use the far right and far left behavior to determine whether the leading coefficient of P is positive or negative. (c) List the zeros of P and for each zero, determine whether it has even multiplicity or odd multiplicity. 3. Use the rational zero theorem to determine the possible rational zeros of P (x) = 21x 10 2x 7 + 7x 4 + 3x Use Descartes Rule of Signs to state the number of possible positive and negative real zeros of the given polynomial functions. (a) P (x) = 2x 5 + 3x 4 + 5x 3 + x 2 + 6x + 5 (b) Q(x) = 6x 6 7x 5 x 4 + 6x 3 5x 2 + 3x Find the zeros of the polynomial P by using synthetic division to divide out one rational zero of P, and then solve the remaining quadratic to find the remaining two zeros P (x) = x x x Find the zeros of the polynomial P by using synthetic division to successively divide out two rational zeros of P and then solve the remaining quadratic to find the remaining two zeros. P (x) = x 4 1x 3 8x 2 + 6x + 12 Page 6

7 7. (a) Find a polynomial P of degree 3 that has zeros 5, 3 and 5. You may leave your answer in factored form. (b) Find a polynomial Q of degree 3 that has zeros 5, 3 and 5 such that Q(0) = 75. You may leave your answer in factored form. 8. (a) Find a polynomial P of degree 3 that has zeros 0, 3 and 3. You may leave your answer in factored form. (b) Find a polynomial Q of degree 3 that has zeros 0, 3 and 3 and Q(1) = 16 You may leave your answer in factored form. Page 7

8 4 The Fundamental Theorem of Algebra 1. Given that 1 + 2i is a zero of Q(x) = x 4 2x 3 + 8x 2 6x + 15 find the remaining zeros, and write Q(x) as a product of linear factors. 2. Find all zeros of the polynomial P defined by given that x = 2i is a zero of P. P (x) = x 4 5x 3 + 5x 2 20x (a) Find a polynomial P with real coefficients of smallest degree that has zeros i, and 2. (b) Find a polynomial P with real coefficients of smallest degree that has zeros i, and 2 such that P (0) = 4. Page 8

9 5 Rational Functions 1. Determine the domain of the rational function Write your answer in interval notation. 2. Consder the rational function F defined by F (x) = x2 + 2x 7 x 2 7x + 6. F (x) = x2 3x + 2 x 2 4x + 3 (a) Find the domain of F ; write your answer in interval notation. (b) Find all x-intercepts of F, if there are any. (c) Find the y-intercept of F, if there is one. (d) Write equations of all vertical asymptotes of F, if there are any. 3. Let f(x) = x 4 1 (x 1) 2 (x + 1) (a) Find the domain of f. (b) Write equations of all vertical asymptotes for the graph of f. (c) For each vertical asymptote x = c, describe the behavior of the graph of f just to the right and just to the left of the asymptote. 4. Let f(x) = x2 8x + 16 x 2 7x (a) State the domain of f, and simplify f if possible. (b) Find equations for the vertical asymptotes for the graph of f. (c) For each vertical asymptote found in part (b), determine the behavior of f just to the right and just to the left of the vertical asymptote. Confirm your answer by creating a table of values. (d) Find all values of c for which there is a hole in the graph of f above x = c. 5. The rational function f(x) = 3x2 7x + 4 is graphed below to the right, and the rational (x 1) function g(x) = x2 + 5x + 6 is graphed below to the left. 3(x + 4) Page 9

10 y f(x) = 3x2 7x + 4 x 1 x y g(x) = x2 + 5x + 6 3(x + 4) x Explain why the function f has a hole in the graph where its denominator is 0, whereas the function g has a vertical asymptote where its denominator is Consider the rational function f(x) = 3x2 10x + 8. x 2 (a) Does f have a vertical asymptote at x = 2? (b) Sketch the graph of f. 7. Find all horizontal asymptotes, or explain why there is no horizontal asymptote, for the rational functions F, G and H defined below. (a) F (x) = 18x8 + 11x x 5 + 5x 11 (b) G(x) = 18x8 + 11x 5 8 5x 8 28x (c) H(x) = x5 + 18x x 5 + 5x 6 8. Find all slant and horizontal asymptotes for the rational function F (x) = 8x3 6x 2 3x + 7 2x Use the rational function F defined below to answer the following questions. F (x) = (x + 2)(x 2) x 3 (a) Find the domain of F. Express answer in interval notation. (b) Find the x-intercept(s) of F, if there are any. (c) Find the y-intercept(s) of F, if there are any. (d) Find the horizontal asymptote(s) of F, if there are any. (e) Find the slant asymptote(s) of F, if there are any. Page 10

11 (f) Find all vertical asymptotes of F, if there are any. For each vertical asymptote, determine the behavior of F just to the right, and just to the left of the asymptote. (g) Use the information from (a) through (f), along with plotting some additional points as necessary to sketch the graph of F along with all of its asymptotes. 10. Let f(x) = x2 10x + 25 x 2 6x + 5. (a) Simplify f and find its domain. (b) Find equations for the vertical asymptote(s) for the graph of f. (c) Find the x- and y-intercepts of the graph of f. (c) For each vertical asymptote found in part (b), determine the behavior of f just to the right and just to the left of the vertical asymptote. (d) Find all values of c for which there is a hole in the graph of f above x = c. (e) Find all horizontal asymptote of f. (f) Use the information above and plot additional points a necessary to graph f. Page 11