Question

If alpha and beta are zeroes of 4x²-x-4, then find the quadratic polynomial whose zeroes are 1/2×alpha and 1/2×beta​

Answer 1

Step-by-step explanation:

α and β are the zeroes of 4x²-x-4

a = 4

b =-1

c = -4

sum of zeroes = -b/a

α + β = -b/a

α + β = -(-1/4)

α + β = 1/4.......................(i)

product of zeroes = c/a

αβ = c/a

αβ = -4/4

αβ = -1 ................(ii)

zeroes of polynomial are 1/2α and 1/2β

so

sum of zeroes =

1/2α + 1/2 β =1/2(1/α +1/ β)

1/2( α + β/ αβ )

1/2(1/4/-1)

-1/8 from (i)

product of zeroes

(1/2α)(1/2β)

1/2(1/α)(1/β)

1/2(1/ αβ)

1/2( 1/-1)

-1/2 from (ii)

polynomial =

x ² - ( sum of zeroes )x + product of zeroes

x² +1/2( α + β/ αβ )x +1/2(1/ αβ)

x² -1/8x-1/2