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

option b is correct

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

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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

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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

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3 12 12 0

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

Answer 2

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

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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

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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 = (2)/(3)`

So zeros are
`-2, (2)/(3)`