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Find all the zeros of the polynomial, $2 x 3 + x 2 − 6 x − 3$ if two of its zeros are $3 and − 3 .$

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− 3, 3

and 2

− 3, 3 and − 12

− 3, 3 and 3

− 3, 3 and − 3

answer is B.

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Detailed Solution

Given that a polynomial

f x = 2 x 3 + x 2 − 6 x − 3

with two of its zeros are

− 3 and 3

.
The zeroes of the polynomial are those values where the entire polynomial is 0.
Given

x = − 3

and

x = 3

are factors of

f x = 2 x 3 + x 2 − 6 x − 3

. Then,

Therefore

x 2 − 3

is also a factor of

f (x)

.
Divide

f x = 2 x 3 + x 2 − 6 x − 3

x 2 − 3

using Division algorithm

⇒ f (x) = (2 x + 1) x 2 − 3

Factorize

(2 x + 1) x 2 − 3 = 0

to find remaining zeroes of

f x

∴ (2 x + 1) x 2 − 3 = 0 ∵ a 2 − b 2 = a + b a − b ⇒ (2 x + 1) x + 3 x − 3 = 0 ⇒ x = − 12, − 3, 3

Therefore, the other zero of

f x

− 12

.
Therefore, all zeroes of the given polynomial

f x = 2 x 3 + x 2 − 6 x − 3

are

− 3, 3 and − 12 .

Hence the correct option is 2.

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