Question

If α and β are the zeroes of a quadratic polynomial 4x 2 + 4x + 1 , then form a quadratic polynomial whose zeroes are 2α and 2β .

Answer 1

If α and β are zeroes, then equation will be

x² - (α+β)x + αβ = 0

Given equation is 4x² +4x + 1=0

∴α +β = -4/4 = -1
αβ = 1/4

Now, new zeroes are 2α and 2β
then,
2α+2β = 2(α+β) = 2(-1) = -2
2α×2β = 4αβ = 4×(1/4) = 1

So the quadratic polynomial will be

x² -(-2)x + 1 =0
⇒x² +2x +1 =0

Answer 2

 equation is 4x² +4x + 1=0

∴α +β = -4/4 = -1
αβ = 1/4

 new zeroes are 2α and 2β
then,
2α+2β = 2(α+β) = 2(-1) = -2
2α×2β = 4αβ = 4×(1/4) = 1

: polynomial will be

x² -(-2)x + 1 =0
⇒x² +2x +1 =0