Question

If alpha and beta are zeros of the polynomial 4 x squared minus x minus 4 find the quadratic polynomial whose zeros are 1 by 2 alpha and 1 by 2 Beta please help me as I need a answer fast

Answer 1

the correct answer is 8x^2-x-2.
since alpha & beta are the roots of the given equation, find alpha+beta & alpha×beta from the equation. then, put these values in the equation whose roots are alpha/2 & beta/2.

Answer 2

Hi friend !

p(x) = 4x² - x - 4
a = 4 , b = -1 , c = -4

α and β are zeros of p(x)

we know that ,

sum of zeros = -b/a
that is ,
α + β = -b/a = 1/4

product of zeros = c/a
that is ,
αβ = -4/4 = -1

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1/2α and 1/2β are zeros of a polynomial

sum of zeros = 1/2α + 1/2β
= 2α + 2β / 4αβ
= 2 [α + β] / 4αβ
= [2 × 1/4] / 4× - 1
= (1/2)/ -4
= -1/8

product of zeros =( 1/2α )( 1/2β)
= 1/4αβ
= 1/4×-1
= -1/4

a quadratic polynomial is given by ,

k {x² - [ sum of zeros ]x + [ product of zeros ]}

k{x² - [-1/8]x - 1/4}

k{x² + 1/8x - 1/4}

putting k = 8

8(x² + 1/8x - 1/4 )


8x² + x - 2 ----> is the required polynomial