2. Approximate the real zero of f(x) = x3 + x + 1 to the nearest tenth. Answer: Substitute all the values into f(x) and find which is closest to zero

Size: px

Start display at page:

Download "2. Approximate the real zero of f(x) = x3 + x + 1 to the nearest tenth. Answer: Substitute all the values into f(x) and find which is closest to zero"

Hugo Barker
6 years ago
Views:

1 Unit 2 Examples(K) 1. Find all the real zeros of the function. Answer: Simply substitute the values given in all the functions and see which option when substituted, all the values go to zero. That is the answer F(-3) is not equal to zero F(1) is not equal to zero This rules out options [A],[B] and [C] Also f(6) = f(3) = f(-1) = 0 Option D is correct 2. Approximate the real zero of f(x) = x3 + x + 1 to the nearest tenth. Answer: Substitute all the values into f(x) and find which is closest to zero F(-1) = -1 F(0.3) = F(-1.3) = = F(-0.7) = = a b. 0.3 c. -1.3

2 d Find all the real zeros of the function. Answer: Simply substitute the values given in all the functions and see which option when substituted, all the values go to zero. That is the answer F(-3) = 0 F(-4) = 0 This leaves us with only [C] 4. Determine the domain of the function. Answer: Domain would be all x except where the denominator would become zero making the function indeterminate. At x= 0 and x= 9 this happens. So [D]

3 5. Identify the equation for the function shown on the graph. Answer: From the graph the roots are 0 and 5, The equation should result in zero when 0 or 5 is substituted. Also the graph tends to infinity as x increases Option [B] and [C] both leads to zero at x = 5. But only option[c] increases to infinity as x increases (Rate of increase of square function is greater than linear function) a. y = x2 + 5x b. y = -x2 + 5x c. y = x2-5x d. y = -x2-5x 6. Simplify INCOMPLETE QUESTION a. b.

4 c. d. 7. Use the Remainder Theorem to find the remainder for the division of (x4-3x2 + 2x - 1) (x - 1). The remainder is. Answer: In remainder theorem, the zero of the divisor (denominator) has to be substituted at the dividend (numerator), The resulting value is the remainder Root of (x-1) is 1 Substitute in x4 3x2 + 2x -1, we get -1 a. -1 b. 1 c. 0 d Determine the maximum number of zeros of the polynomial function. Answer: Maximum number of zeros is equal to the degree of the polynomial, which is 8

5 9. Write the quotient in standard form. Multiply and divide by (7 7i) Numerator: (-7-i) *(7-7i) = i-7i-7 = i = -14(4-3i) Denominator: (-7-7i)*(7-7i) = -1( ) = -14 * 7 Nr / Dr = (4-3i)/7 10. Question: Answer: To solve this MCQ, jus substitute the roots of all the given options in the main polynomial and find out which one doesn t go to zero. To do it systematically, you ll have to do long division. By just observation, any positive value substituted would never make the dividend goto zero. But, root of option[d] is positive. So, [D] is the answer

6 11. Identify the graph of the rational function. Find any vertical and horizontal asymptotes. Answer: As x approaches 1, denominator diminishes thereby shooting up the absolute value of the function. As it approached from right side, the function is positive and reached positive infinity, whereas when approached from left side, the function is negative and goes to negative infinity. This leaves us with only [A] as it is very clear that x=1 is an asymptote

7 12. Question: Answer: [B] and [C] ae representation-wise correct, But the root has to be between the test points. Only 2.41 is between 2 and 3. So [C] is the answer 13. Question: Answer: Complex zeroes appear in pairs. So +5i and -5-i are also zeroes

8 14. What is the y-intercept of the function y = 2x2 + 1 Answer: At y intercept, x is zero Y(x = 0) = 0 +1 = 1 a. 3 b. 1 c. 0 d. none 15. Graph f(x) = x3-4x2 + 2x 3. Answer: F(1) = -4 is not zero F(2) = -7 is not zero F(3) = -6 is not zero F(4) = 5 is not zero But there is a sign change between f(3) and f(4) indicating presence of zero between 3 and 4 Option[B} explains all the calculations a.

9 b. c. d. 16. What is the equation of the vertical asymptote of the function? FUNCTION NOT GIVEN a. x = -4 b. y = 1 c. x = 2 d. x = 1

10 17. Question: Answer: Find the remainder using remainder theorem X = -2 is root of the denominator Substituting in numerator, we get, -31 In option[a], -31 is remainder, So it is correct 18. Determine the domain of the function. Answer: Domain would be all x except where the denominator would become zero making the function indeterminate. At x= 0, x= 4 and x= -4 this happens. So [B]

11 19. Find the vertex of the graph of f(x) = x2 + 4x - 5. Vertex is the root of the derivative of the given function G(x) = Df/dx = 2x + 4 G(-2) = is root of derivative F(-2) = -9 Vertex (-2,-9) a. (0, -5) b. (-2, -9) c. (-2, 9) d. (2, 7) 20. Find the vertex of the graph of the function f(x) = x2-6x + 5. Vertex is the root of the derivative of the given function G(x) = Df/dx = 2x - 6 G(3) = 0. 3 is root of derivative F(3) = -4 Vertex (3,-4) a. (3, 4) b. (-6, 5) c. (3, -4) d. (-3, 4)

12 21. Question: Answer: Complex zeroes appear in pairs. -2-3i is also a root 22. Determine the maximum number of zeros of the polynomial function. Answer: Maximum zeroes would be equal to the degree, which is 2 a. 2 b. 4 c. 3 d Which of the following graphs best represents the function y = x2 + 7x + 8? Answer: F(0) = 8 ALSO THE GRAPH DECREASES AS X INCREASES So [B] fits the criteria

13 a. b. c. d.

14 24. Identify the graph of the rational function. Find any vertical and horizontal asymptotes. Answer: F(-2) = F(3) = 0 So x=-2 and x=3 are asymptotes Only [B] lists that out correctly

15 25. Question: Answer: (-8 + 4i)(5-7i) = i + 20i 28(i2) = 76i 40 28*(-1) = 76i (40-28) = i 26. Question: Answer, As x reaches infinity, denominator would increase tremendously and the function would approach zero. So, y=0 is asymptote 27. Use the Quadratic Formula to solve.

16 Answer: Direct application of formula 28. What is the conjugate of 12-4i? a. -4i b i c. d Question: 30. Divide x4 + 2x2-1 by x - 1 using synthetic division. The result of synthetic division is.

17 a b c d Simplify -8i 6i. a. 48 b. -48 c. d One of the factors of x3-7x + 6 is x - 1. Which of the following is one of the other factors? Answer: Substitute the roots of the given options, the option for which the given function amounts to zero is the answer F(2) = 0 2 is the root of option[d]

18 a. x + 1 b. x - 6 c. x - 3 d. x Wite the quotient in standard form. Multiply and divide by -3i Numerator: (3-4i) *(-3i) = -9i 12 = 3 (-3i - 4) Denominator: 9 = 3 *3 Nr / Dr = (-4-3i)/3

19 34. Question: Answer C(x=0) = (0+150) / (5(0+150)) = 1/5 = 0.2 Only [B] and [D] satisfies this condition As x approaches infinity, C tends to just the ratio of co-efficient of x in numerator and denominator As x-> infinity, C = 4/5 = 0.8 Option[D] is correct

20 35. Question: Answer: Upon applying remainder theorem to each of the given factors, only in option [C] all the remainders came out to be zero. SO option [C] is correct 36. Write the partial fraction decomposition of Answer: Reconstructing a single fraction out of option[b], we get Numerator: 2(x+2) + 3(x-1) = 2x + 3x = 5x + 1 Denominator: (x-1)(x+2) = x2 +x -2 So, option [B] is correct a. b.

21 c. d. Answer not given 37. Factor 2x3+x2-5x+2 given that x+2 is one of its factors. Answer: Upon applying remainder theorem to each of the given factors, only in option [A] all the remainders came out to be zero. SO option [A] is correct a. (x+2)(x-1)(2x-1) b. (x+2)(x+1)(2x+1) c. (x+2)(x+1)(2x-1) d. (x+2)(x-1)(2x+1) 38. Question: Answer: 7i + 3i2 = 7i + 3(-1) = i

22 39. How many possible negative real zeros are there for the polynomial g(x) = 2x4 + x3-4x + 1? Answer: As evident from the graph, the given function has no real roots ANSWER SHOULD BE 0 a. 1 b. 3 c. 5 d. 2

23 40. Question: Answer: Vertical asymptotes happens at the value of x co-ordinate where the denominator goes to zero. At x=2 and x=3, the denominator goes to 0 and hence they are the asymptotes (vertical) 41. Use synthetic division to determine how many of the given factors divide the polynomial evenly. Answer: Direct method would be to use synthetic division, but to solve this MCQ, we can use remainder theorem for each and every factor given and find out the number of factors for which the result was not zero Let the polynomial be f F(x=4) = 0 F(x=-4) = -224 F(x=(2/3)) = 0 F(x = -1) = 25 Two of the given factor leave zero remainder and hence divide the polynomial evenly. a. Three of the factors divide the polynomial evenly.

24 b. All four of the factors divide the polynomial evenly. c. Two of the factors divide the polynomial evenly. d. One of the factors divides the polynomial evenly. 42. Question: Answer: As x goes to negative infinity, x7 goes to a very high negative value, but the co-efficient is negative, making the graph increase As x goes to infinity, x7 value increases tremendously, but the co-eff value is negative, thereby decreasing the graph. So Rises to the left, Falls to the right a. Rises to the left. Falls to the right. b. Rises to the left. Rises to the right. c. Falls to the left. Falls to the right. d. Falls to the left. Rises to the right. 43. List all of the possible rational zeros of the function. FUNCTION NOT PROVIDED a. b. c. d. 44. Which function has a graph that opens upward?

25 Answer: A positive co-efficient for the second degree term in a quadratic polynomial, lets the graph open upward. Only [C] satisfies a. f(x) = -x2 + 4x + 7 b. f(x) = -x2 + 3x -12 c. f(x) = 2x2-5x - 2 d. f(x) = -5x2-6x Give the domain of the given function. Answer: Domain would be all x except where the denominator would become zero making the function indeterminate. At x= 2 and x= -5 this happens. So [A] a. All real numbers except 2 and -5 b. All real numbers except -2 and -5. c. All real numbers except -2 and 5. d. Answer not given 46. If a = 2 and the vertex is (3, -4), will there be a minimum or maximum value for the parabola? What is the value? Answer: Positive quadratic co-efficient, graph opens upward. Implies the vertex will be minimum with a value of -4 (given) a. maximum, -3 b. maximum, 3 c. maximum, -4 d. minimum, -4

26 47. Question: Answer: 3 roots given, so a cubic polynomial. Only [C] and [D] satisfy. Also as x tends to infinity the graph increases, this implies a positive co-efficient for x^3. So answer is [C] 48. Simplify (8-9i) + (3 + 4i). Answer: (8-9i)+(3+4i) = (8+3) + (4i-9i) = 11 5i a. -i + 17 b i c. 11-5i

27 d i 49. Question: Answer: The graph has even power, so the graph increases/decreases both sides. [B] and [D] fail F(2) = = 3, [C] fails to meet this criteria So, [A] is correct

28 50. Question: Answer:

Chapter 2 Formulas and Definitions:

Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)