Q31. Use the Intermediate Value Theorem to determine whether the polynomial func

Question

Q31. Use the Intermediate Value Theorem to determine whether the polynomial function f(x) = -4x4 - 9x2 + 4; has a zero in the interval [-1, 0].
   a. f(-1) = -9 and f(0) = -4; no
   b. f(-1) = 9 and f(0) = 5; no
   c. f(-1) = 9 and f(0) = -4; yes
   d. f(-1) = -9 and f(0) = 4; yes

Q32. Find the domain of the rational function f(x) = (x + 9)/(x2 - 4x).
   a. {x|x -2, x 2}
   b. {x|x -2, x 2, x -9}
   c. all real numbers
   d. {x|x 0, x 4}

Q33. Use the Factor Theorem to determine whether x + 5 is a factor of f(x) = 3x3 + 13x2 - 9x + 5.
   a. Yes
   b. No

Q35. For the polynomial f(x) = (1/5)x(x2 - 5), list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept.
   a. 0, multiplicity 1, touches x-axis; 5, multiplicity 1, touches x-axis; -5, multiplicity 1, touches x-axis
   b. 0, multiplicity 1, crosses x-axis; 5, multiplicity 1, crosses x-axis; -5, multiplicity 1, crosses x-axis
   c. 5, multiplicity 1, touches x-axis; -5, multiplicity 1, touches x-axis
   d. 0, multiplicity 1

Q37. Find the x-intercepts of the graph of the function f(x) = (x - 3)(2x + 7)/(x2 + 9x - 8).
   a. (3, 0), (-7, 0)
   b. (-3, 0), (7/2, 0)
   c. (3, 0), (-7/2, 0)
   d. none

Q38. Find the power function that the graph of f(x) = (x + 4)2 resembles for large values of |x|.
   a. y = x8
   b. y = x2
   c. y = x4
   d. y = x16

Q39. Find the domain of the rational function R(x) = (-3x2)/(x2 + 2x - 15).
   a. {x|x 5, 3}
   b. {x|x 5, -3}
   c. {x|x - 15, 1}
   d. {x|x -5, 3}

Q40. The function f(x) = x4 - 5x2 - 36 has the zero -2i. Find the remaining zeros of the function.
   a. 2i, 6, -6
   b. 2i, 3i, -3i
   c. 2i, 3, -3
   d. 2i, 6i, -6i

Explanation / Answer

31. Let f(x) = -4x4 - 9x2 + 4. Then f(-1) = -4*1 -9*1 +4 = -9. Also, f(0) = 4. Then, as per the Intermediate Value Theorem, f(x) = 0 at some point in the interval [-1,0], i.e. the given polynomial has a root between -1 and 0. Option (c ) is the correct answer.

32. The denominator of the given function is x2 -4x = x(x-4) which is 0 when x = 0 or, x = 4. Since the division by 0 is not defined, the domain of the given rational function is { x R| x0 , x4}. Option (d) is the correct answer.

33. As per the factor theorem, x +5 is a factor of f(x) = 3x3 + 13x2 - 9x + 5 if f(-5) = 0. Now, f(-5) =             3* (-125) +13*25 -9*(-5) +5 = -375 + 325+45 +5 = 0. Hence x+5 is a factor of f(x). The answer is yes.

35. The zeros of the polynomial f(x) = (x/5)(x2 - 5) are where x/5 = 0 or, x2 – 5 = 0 i.e. 0 and ±5. All these 3 zeros are of multiplicity 1. Hence the graph crosses the x-axis at each x-intercept. Option (b) is the correct answer.

37. We know that the x-intercept is where y = 0. Hence, for f(x) = (x - 3)(2x + 7)/(x2 + 9x - 8), the x-intercepts are given by x -3 = 0, i.e. x = 3 and 2x+7 = 0, i.e. x = -7/2. Option (c ) is the correct answer.

38. For large values of |x|, the power function that the graph of f(x) = (x + 4)2 resembles is x4. Option (c ) is the correct answer.

39. Since division by 0 is not defined, and since x2 + 2x – 15 = (x+5)(x-3) = 0 when x = -5 or x = 3, hence the domain of f(x) is { x R| x-5 , x3}. Option (d) is the correct answer.

40. We know that complex roots occur in conjugate pairs. Hence if -2i is a zero of f(x) = x4 - 5x2 – 36, then +2i is also a zero. Thus (x+2i)(x-2i) = (x2+4) is a factor of f(x). Then f(x) = x4 - 5x2 – 36 = (x2 -9)(x2+4) so that the other zeros of f(x), being zeros of (x2-9) are -3 and 3. Option (c ) is the correct answer.

Related Questions

Navigate

• Browse (All)
• Browse Q
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.