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if $α$ and $β$ are the zeros of the quadratic polynomial $f x = x 2 − 2 x + 3$ find a polynomial whose roots are $α − 1 α + 1, β − 1 β + 1 .$

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k x 2 − 23 x + 13

k x 2 − 23 x − 13

k x 2 + 23 x + 13

k x 2 + 23 x − 13

answer is A.

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Detailed Solution

Given that

α

and

β

are the zeroes of the quadratic polynomial

f x = x 2 − 2 x + 3

.
Quadratic equation can be represented by,

p (x) = k x 2 − (sum of zeroes) x + (product of zeroes)

, where

k

is any non-zero number.
A polynomial whose roots are

α − 1 α + 1 and β − 1 β + 1

.
Compare the given polynomial

f x = x 2 − 2 x + 3

with general polynomial

f (x) = a x 2 + b x + c .

a = 1, b = − 2

and

c = 3

Sumof zeros = − b a α + β = − (− 2) 1 = 2

Product of zeros = c a α β = 31 = 3

Let the new polynomial be

g x

with roots are

α − 1 α + 1 and β − 1 β + 1

Substitute

α + β = 2 and α β = 3.

∴ Sumof zeros = 2 α β − 2 α β + (α + β) + 1 = 2 (3) − 2 3 + 2 + 1 = 46 = 23

Product of zeros = α − 1 α + 1 × β − 1 β + 1 = (α − 1) (β − 1) (α + 1) (β + 1) = α β − α − β + 1 α β + α + β + 1 = α β − (α + β) + 1 α β + (α + β) + 1

Substitute

α + β = 2 and α β = 3.

∴ Product of zeros = α β − (α + β) + 1 α β + (α + β) + 1 = 3 − 2 + 1 3 + 2 + 1 = 26 = 13

Substitute sum of zeroes

= 23

and product of zeroes

= 13

into the polynomial

p (x) = k x 2 − (sum of zeroes) x + (product of zeroes),

∴ p (x) = k x 2 − (sumof zeroes) × + product of zeroes = k x 2 − 23 x + 13 = k x 2 − 23 x + 13

The polynomial whose roots are

α − 1 α + 1, β − 1 β + 1

k x 2 − 23 x + 13 .

Hence the correct option is 1.

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