# If α And β Are the Zeros of the Quadratic Polynomial F(X) = X2 − 1, Find a Quadratic Polynomial Whose Zeroes Are (2alpha)/Beta" and "(2beta)/Alpha - Mathematics

If α and β are the zeros of the quadratic polynomial f(x) = x2 − 1, find a quadratic polynomial whose zeroes are (2alpha)/beta" and "(2beta)/alpha

#### Solution

Since α and β are the zeros of the quadratic polynomial f(x) = x2 − 1

The roots are α and β

alpha+beta="-coefficient of x"/("coefficient of "x^2)

alpha+beta=0/1

alpha+beta=0

alphabeta="constant term"/("coefficient of "x^2)

alphabeta=(-1)/1

alphabeta=-1

Let S and P denote respectively the sum and product of zeros of the required polynomial. Then,

S=(2alpha)/beta+(2beta)/alpha

Taking least common factor we get,

S=(2alpha^2+2beta^2)/(alphabeta)

S=(2(alpha^2+beta^2))/(alphabeta)

S=(2[(alpha+beta)-2alphabeta])/(alphabeta)

S=(2[(0)-2(-1)])/-1

S=(2[-2(-1)])/-1

S=(2xx2)/-1

S=4/-1

S = -4

P=(2alpha)/betaxx(2beta)/alpha

P = 4

Hence, the required polynomial f(x) is given by,

f(x) = k(x2 - Sx + P)

f(x) = k(x2 -(-4)x + 4)

f(x) = k(x2 +4x +4)

Hence, required equation is f(x) = k(x2 +4x +4) Where k is any non zero real number.

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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 2 Polynomials
Exercise 2.1 | Q 15 | Page 35