Question

Use synthetic division to determine which of the following are the real zeroes of the function f left parenthesis x right parenthesis equals 3 x cubed plus 10 x squared plus 4 x minus 8.
(1 point)
Responses

negative 3 halves comma 2
Image with alt text: negative 3 halves comma 2

negative 2 comma 2 thirds
Image with alt text: negative 2 comma 2 thirds

negative 2 comma negative 2 thirds comma 2
Image with alt text: negative 2 comma negative 2 thirds comma 2

negative 2 comma 3 halves comma 2

Answer 1

is 2 a real zero

Answer 2

To use synthetic division, we need to find the zeroes of the function. To do this, we will set the function equal to zero and solve for x:

3x^3 + 10x^2 + 4x - 8 = 0

Now, we can use synthetic division with the possible zeroes given to determine which ones are real zeroes.

For the first option, x = -3/2:

-3/2 | 3 10 4 -8
-9 4 -12
-------------
3 1 -8

The resulting polynomial is 3x^2 + x - 8. Since the remainder is not zero, -3/2 is not a real zero.

For the second option, x = -2/3:

-2/3 | 3 10 4 -8
-2 4 -8
-------------
3 8 -4

The resulting polynomial is 3x^2 + 8x - 4. Since the remainder is not zero, -2/3 is not a real zero.

For the third option, x = -2:

-2 | 3 10 4 -8
-6 -8 8
------------
3 4 0

The resulting polynomial is 3x^2 + 4x, which has a remainder of 0. Therefore, -2 is a real zero.

For the fourth option, x = 3/2:

3/2 | 3 10 4 -8
9/2 27/2 33/2
----------------
3 17/2 25/2

The resulting polynomial is 3x^2 + (17/2)x + (25/2), which has a remainder of 0. Therefore, 3/2 is a real zero.

So, the real zeroes of the function f(x) = 3x^3 + 10x^2 + 4x - 8 are -2 and 3/2.