Question

if alpha and beta are the zeros of the quadratic polynomial f(x) = x^2 - 1 , find a quadratic polynomial whose zeroes are 2α/β and 2β/α ​

Answer 1

Step-by-step explanation:

f(x)=x²-1

Roots are Alpha ,beta

Alpha +Beta=-b/a=0

Alpha×beta=c/a=-1

Now we have to Find Polynomial whose Root are 2alpha/beta,2beta/alpha

Now we Structure of Quadratic Equation

ax²+(sum of Roots)x+product of Roots

So sum of Roots

2alpha/beta+2beta/alpha

2alpha²+2beta²/alpha×beta

(For Convience i m taking alpha=x,beta=y)

2x²+2y²/xy

2(x²+y²)/xy

We know

(x+y)²-2xy=x²+y²

0²-2×-1=x²+y²

x²+y²=2

xy=-1

Sum of Roots 4/-1=-4

Product of Roots

4xy/xy

=4

Sum=-4

product=4

Quadratic Equation

x²+4x+4

p(x)=x²+4x+4

Answer 2

p(x)=x raiseto power2 +4x+4