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If two zeroes of the polynomial $p (x) = x 4 + x 3 − 9 x 2 − 3 x + 18$ are $3$ and $− 3$ , find the remaining zeroes.

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1 and -3

1 and 3

-3 and 2

-3 and 1

answer is B.

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Detailed Solution

Given polynomial

p (x) = x 4 + x 3 − 9 x 2 − 3 x + 18

has two of its zeroes as

3

and

− 3

. 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

3

and

− 3

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

x − 3

and

x + 3 .

Calculating the product of these factors we have,

x − 3 x + 3 = x 2 − 3

Now dividing the given polynomial

x 4 + x 3 − 9 x 2 − 3 x + 18

x 2 − 3

we have,

Quotient obtained is

x 2 + x − 6

.
Factorizing the quotient obtained we have,

x 2 + x − 6 = 0 ⇒ x 2 + 3 x − 2 x − 6 = 0 ⇒ x x + 3 − 2 x + 3 = 0 ⇒ x + 3 x − 2 = 0 ∴ x = − 3, 2

Hence, the remaining zeros of the polynomial are

2

and

− 3

.
Therefore, option 2 is correct.

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