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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`

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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
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