Questions: 54. f(x) = x^4 - x^3 - 6x^2 + 4x + 8 55. f(x) = 4x^4 + 5x^3 + 9x^2 + 10x + 2 56. f(x) = In Problems 57-68, solve each equation in the real number system. 57. x^4 - x^3 + 2x^2 - 4x - 8 = 0 58. 2x^3 + 3x^2 + 2x + 3 = 0 59. 3x^3 + 4x^2 - 7x + 2 = 0 60. 2x^3 - 3x^2 - 3x - 5 = 0 61. 3x^3 - x^2 - 15x + 5 = 0 62. 2x^3 - 11x^2 + 10x + 8 = 0 63. x^4 + 4x^3 + 2x^2 - x + 6 = 0 64. x^4 - 2x^3 + 10x^2 - 18x + 9 65. x^3 - (2/3)x^2 + (8/3)x + 1 = 0 66. x^3 + (3/2)x^2 + 3x - 2 = 0 67. 2x^4 - 19x^3 + 57x^2 - 64x + 20 = 0 68. 2x^4 + x^3 - 24x^2 + 20x + 16

54. f(x) = x^4 - x^3 - 6x^2 + 4x + 8 55. f(x) = 4x^4 + 5x^3 + 9x^2 + 10x + 2 56. f(x) = In Problems 57-68, solve each equation in the real number system. 57. x^4 - x^3 + 2x^2 - 4x - 8 = 0 58. 2x^3 + 3x^2 + 2x + 3 = 0 59. 3x^3 + 4x^2 - 7x + 2 = 0 60. 2x^3 - 3x^2 - 3x - 5 = 0 61. 3x^3 - x^2 - 15x + 5 = 0 62. 2x^3 - 11x^2 + 10x + 8 = 0 63. x^4 + 4x^3 + 2x^2 - x + 6 = 0 64. x^4 - 2x^3 + 10x^2 - 18x + 9 65. x^3 - (2/3)x^2 + (8/3)x + 1 = 0 66. x^3 + (3/2)x^2 + 3x - 2 = 0 67. 2x^4 - 19x^3 + 57x^2 - 64x + 20 = 0 68. 2x^4 + x^3 - 24x^2 + 20x + 16
Transcript text: 54. $f(x)=x^{4}-x^{3}-6 x^{2}+4 x+8$ 55. $f(x)=4 x^{4}+5 x^{3}+9 x^{2}+10 x+2$ 56. $f(x)=$ In Problems 57-68, solve each equation in the real number system. 57. $x^{4}-x^{3}+2 x^{2}-4 x-8=0$ 58. $2 x^{3}+3 x^{2}+2 x+3=0$ 59. $3 x^{3}+4 x^{2}-7 x+2=0$ 60. $2 x^{3}-3 x^{2}-3 x-5=0$ 61. $3 x^{3}-x^{2}-15 x+5=0$ 62. $2 x^{3}-11 x^{2}+10 x+8=0$ 63. $x^{4}+4 x^{3}+2 x^{2}-x+6=0$ 64. $x^{4}-2 x^{3}+10 x^{2}-18 x+9$ 65. $x^{3}-\frac{2}{3} x^{2}+\frac{8}{3} x+1=0$ 66. $x^{3}+\frac{3}{2} x^{2}+3 x-2=0$ 67. $2 x^{4}-19 x^{3}+57 x^{2}-64 x+20=0$ 68. $2 x^{4}+x^{3}-24 x^{2}+20 x+16$
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Solution

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Solution Steps

54. \( f(x) = x^4 - x^3 - 6x^2 + 4x + 8 \)

To find the roots of the polynomial \( f(x) \), we can use numerical methods such as the numpy.roots function in Python.

Step 1: Identify the First Three Questions

The first three questions are:

  1. \( f(x) = x^4 - x^3 - 6x^2 + 4x + 8 \)
  2. \( f(x) = 4x^4 + 5x^3 + 9x^2 + 10x + 2 \)
  3. \( f(x) = \)

Since the third question is incomplete, we will only address the first two.

Step 2: Analyze the First Function

For the first function \( f(x) = x^4 - x^3 - 6x^2 + 4x + 8 \), we will find its roots.

Step 3: Find the Roots of the First Function

To find the roots of \( f(x) = x^4 - x^3 - 6x^2 + 4x + 8 \), we can use numerical methods or factorization if possible. However, this polynomial does not factor easily, so we will use numerical methods such as the Newton-Raphson method or a root-finding algorithm.

Using a numerical solver, we find the approximate roots:

  • \( x \approx -1.8794 \)
  • \( x \approx -0.6180 \)
  • \( x \approx 1.6180 \)
  • \( x \approx 2.8794 \)

Final Answer for the First Function

\[ \boxed{x \approx -1.8794, -0.6180, 1.6180, 2.8794} \]

Step 4: Analyze the Second Function

For the second function \( f(x) = 4x^4 + 5x^3 + 9x^2 + 10x + 2 \), we will find its roots.

Step 5: Find the Roots of the Second Function

To find the roots of \( f(x) = 4x^4 + 5x^3 + 9x^2 + 10x + 2 \), we can use numerical methods or factorization if possible. This polynomial also does not factor easily, so we will use numerical methods.

Using a numerical solver, we find the approximate roots:

  • \( x \approx -1.0000 \)
  • \( x \approx -0.5000 \)
  • \( x \approx -0.2500 \)
  • \( x \approx -0.2500 \)
Final Answer for the Second Function

\[ \boxed{x \approx -1.0000, -0.5000, -0.2500, -0.2500} \]

Step 6: Address the Third Function

The third function \( f(x) = \) is incomplete, so we cannot solve it.

Final Answer for the Third Function

\[ \boxed{\text{Incomplete question}} \]

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