Question

Use synthetic division to determine which of the following are the real zeroes of the function f(x) = 3x^3 + 10x^2 + 4x - 8

Answer 1

Answer : option B

$f(x) = 3x^3 + 10x^2 + 4x - 8$

We have -2 in the given option

Lets use synthetic division and divide by -2

First write the coefficient of all the terms in f(x)

-2           3             10             4              -8

             0              -6            -8              8

         ------------------------------------------------------

           3               4             -4               0

We got remainder =0 , so -2 is one of the zero

Resultant  expression is 3x^2 + 4x -4

Lets use synthetic division and divide by -2 again

-2           3             4             -4            

             0             -6             4

         ------------------------------------------------------

           3               -2             0

We got remainder =0 , so -2 is again a zero

Resultant  expression is 3x - 2

we set 3x -2 =0

Add 2 on both sides

3x = 2

Divide by 3 on both sides

$x = \frac{2}{3}$

So zeros are $-2, \frac{2}{3}$

Answer 2

Answer:

option b is correct

Step-by-step explanation:

In the options a, c and d we have to check if 2 is a root.

The coefficients of  f(x) = 3x^3 + 10x^2 + 4x - 8 are: 3, 10, 4 and -8.

The division of f(x) by (x-2) using synthetic division is:

2          3             10             4              -8

                           6             32              72

        ------------------------------------------------------

           3             16            36              64

The remainder is 64, so 2 is not root of f(x); if it was the remainder would be zero. This result makes options a, c and d incorrect.

The division of f(x) by (x+2) using synthetic division is:

-2          3             10             4              -8

                           -6             -8              8

        ------------------------------------------------------

            3               4            -4               0

The remainder is 0, so -2 is root of f(x).

The division of f(x) by (x-2/3) using synthetic division is:

2/3        3             10             4              -8

                           2               8               8

        ------------------------------------------------------

            3               12           12             0

The remainder is 0, so 2/3 is root of f(x). Then, option b is correct