Find all Zeros of a function: f(x)= 3x^4 - x^3 + 3x^2 + 29x -10

[supremacy32]

Thanks for taking a look. I have a basic understanding of the material and was hoping to have someone doublecheck my work, and let me know if/where I went wrong.

Problem is as follows:

For the polynomial function f(x)= 3x⁴ - x³ + 3x² + 29x -10

a) Find all possible rational zeros:
b) List the zeros of the function:

Solution:

a) Possible zeroes include -1, 1, -2, 2, -5, 5, -10, 10, -1/3, 1/3, -2/3, 2/3, -5/3, 5/3, -10/3, 10/3
b) Zeroes of the Function:

Using synthetic division and guessing and checking determine that 1/3 is a zero, making the equation f(x)=(x-(1/3))(3x³+3x+30)
Using synthetic division determine that -2 is now a zero, making the equation f(x)= (x-(1/3))(x+2)(3x²-5x+15)
Use quadratic equation to determine that remaining zeros are (5/6) ± (√155/6)i
Thus the zeros are 1/3, -2, and (5/6) ± (√155/6)i

On step b, I feel good until the zero that is a complex number. If someone would double check that part especially I would be grateful.

 

[Deleted member 4993]

Thanks for taking a look. I have a basic understanding of the material and was hoping to have someone doublecheck my work, and let me know if/where I went wrong.

Problem is as follows:

For the polynomial function f(x)= 3x⁴ - x³ + 3x² + 29x -10

a) Find all possible rational zeros:
b) List the zeros of the function:

Solution:

a) Possible zeroes include -1, 1, -2, 2, -5, 5, -10, 10, -1/3, 1/3, -2/3, 2/3, -5/3, 5/3, -10/3, 10/3
b) Zeroes of the Function:

Using synthetic division and guessing and checking determine that 1/3 is a zero, making the equation f(x)=(x-(1/3))(3x³+3x+30)
Using synthetic division determine that -2 is now a zero, making the equation f(x)= (x-(1/3))(x+2)(3x²-5x+15)
Use quadratic equation to determine that remaining zeros are (5/6) ± (√155/6)i
Thus the zeros are 1/3, -2, and (5/6) ± (√155/6)i

On step b, I feel good until the zero that is a complex number. If someone would double check that part especially I would be grateful.

y = x² + 2 has a complex zero.

It is quite reasonable....

 

[Ishuda]

Thanks for taking a look. I have a basic understanding of the material and was hoping to have someone doublecheck my work, and let me know if/where I went wrong.

Problem is as follows:

For the polynomial function f(x)= 3x⁴ - x³ + 3x² + 29x -10

a) Find all possible rational zeros:
b) List the zeros of the function:

Solution:

a) Possible zeroes include -1, 1, -2, 2, -5, 5, -10, 10, -1/3, 1/3, -2/3, 2/3, -5/3, 5/3, -10/3, 10/3
b) Zeroes of the Function:

Using synthetic division and guessing and checking determine that 1/3 is a zero, making the equation f(x)=(x-(1/3))(3x³+3x+30)
Using synthetic division determine that -2 is now a zero, making the equation f(x)= (x-(1/3))(x+2)(3x²-5x+15)
Use quadratic equation to determine that remaining zeros are (5/6) ± (√155/6)i
Thus the zeros are 1/3, -2, and (5/6) ± (√155/6)i

On step b, I feel good until the zero that is a complex number. If someone would double check that part especially I would be grateful.

It should be
(x-(1/3))(x+2)(3x²-6x+15) = (3x-1)(x+2)(x²-2x+5)=0