Question

if alpha and beta are the zeros of the polynomial 4x^2-x-4,find the quadratic polynomial whose zeros are 2/alpha and 2/beta​

Answer 1

Answer:

o after dividing by 4 we have

4

4x

2

+4x+1

Step-by-step explanation:

2

+4x+1

x

2

+x+

4

1

Answer 2

Answer:

$2 {x}^{2} + x - 8$

Step-by-step explanation:

For the given polynomial -

Alpha + beta = -(-1)/4 (Sum of the roots)

=> Alpha + beta = 1/4 (1)

Similarly,

Alpha × Beta = -4/4 = -1 (2) (Product of the roots)

Now for the polynomial for which 2/alpha and 2/beta are roots -

2/alpha + 2/beta (Sum of the roots)

=> (2 alpha + 2 beta)/alpha × beta

=> 2(alpha + beta)/ alpha × beta (3)

From (1) and (2) and (3)

=> [2 × (1/4)] / -1

=> -1/2

-1/2

Similarly,

2/Alpha × 2/beta (Product of the roots)

=> 4/ Alpha × beta

Now from (2)

=> 4/-1 = -4/1

Polynomial -

K [ x^2 - (Sum of the roots) x + (Product of the roots) ]

= K [x^2 - (-1/2) x + (-4) ]

( here 'K' is a constant, let K = 2 )

= 2 ( x^2 + 1/2 x - 4 )

= 2x^2 + x - 8

Hence the required polynomial is -

$2 {x}^{2} + x - 8$

* Roots and Zeros are one and the same things