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Form a quadratic polynomial one of whose zeroes is $2 + 5$ and the sum of the zeroes is $4$ .

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2 x 2 − 4 x − 1

x 2 + 4 x + 1

2 x 2 + 4 x + 1

x 2 − 4 x − 1

answer is D.

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Detailed Solution

Given that one of the zeroes of a polynomial is

2 + 5

. The sum of the zeroes of that polynomial is

4

. We need to find the quadratic polynomial.
A quadratic polynomial with zeroes

α

and

β

is given as

x − α x − β = 0 ... (i)

We have,

α = 2 + 5

α + β = 4 ... (i i)

Equating the value of

α

in equation

i i

, we have,

2 + 5 + β = 4 ⇒ β = 4 − 2 − 5 ⇒ β = 2 − 5

Now equating the value of

α

and

β

in equation

i

, we have,

x − 2 + 5 x − 2 − 5 = 0 ⇒ x 2 − 2 + 5 x − 2 − 5 x + 22 − 52 = 0 ⇒ x 2 − 4 x − 1 = 0

The quadratic polynomial is

x 2 − 4 x − 1

with zero

2 + 5

and sum of zeroes 4 .
Therefore, option 4 is correct.

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