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Find the remaining zeroes of polynomial $p (x) = 3 x 4 + 6 x 3 − 2 x 2 − 10 x − 5$ if two of its zeroes are $− 1, − 1.$

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53 and − 53

− 53 and − 53

35 and − 35

− 35 and − 35

answer is A.

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Detailed Solution

Given polynomial

3 x 4 + 6 x 3 − 2 x 2 − 10 x − 5

has two of its zeroes as

− 1

and

− 1.

We need to find the remaining zeroes.
By factor theorem, we have that if

a

is the zero of a polynomial, then

(x − a)

is the factor of that polynomial.
Since

− 1

and

− 1

are the two zeros of the polynomial, therefore, the respective factors are

x + 1

and

x + 1.

Calculating the product of these factors we have,

x + 1 x + 1 = x 2 + x + x + 1 ⇒ x + 1 x + 1 = x 2 + 2 x + 1

Now dividing the given polynomial

3 x 4 + 6 x 3 − 2 x 2 − 10 x − 5

x 2 + 2 x + 1

we have,

Quotient obtained is

3 x 2 − 5.

Equating the quotient to

0,

we have,

3 x 2 − 5 = 0 ⇒ 3 x 2 = 5 ⇒ x 2 = 53

⇒ x = ± 53 ∴ x = 53, − 53

The remaining zeroes of the given polynomial are:

53, − 53

.
Therefore, option 1 is correct.

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