Question

If a and B are the zeroes of the quadratic polynomial f(x) = x² - 5x + k such that a- B = 1, find quadratic polynomial whose zeroes are alpha²/beta and beta²/ alpha​

Answer 1

Step-by-step explanation:

Given:-

a and B are the zeroes of the quadratic polynomial f(x) = x² - 5x + k such that a- B = 1

To find:-

Find quadratic polynomial whose zeroes are alpha²/beta and beta²/ alpha

Solution:-

Given that

The Quadratic Polynomial :f(x)= x² - 5x + k

On Comparing this with the standard quadratic Polynomial ax^2+bx+c

a = 1

b=-5

c=k

Given that a and B are the zeores then

Sum of the zeroes = a+B = -b/a

a+B = -(-5)/1

a+B= 5----------(1)

Product of the zeroes = aB = c/a

aB = k/1

aB = k --------(2)

and given that

a-B = 1-------(3)

On solving (1) &(3)

a+B = 5

a-B = 1

(+)

________

2a +0 = 6

________

=> 2a = 6

=> a = 6/2

=> a = 3

now from (3)

3-B = 1

=> B=3-1

B= 2

aB = k

=> k = 2×3 = 6

The polynomial is x^2-5x+6

The zeroes 2 and 3

α + β = 2+3 = 5

α β = 6

α^2/ β= 2^2/3

=> 4/3

β^2/α=3^2/2

=> 9/2

We know that

If α and β are the zeroes them the Quadratic Polynomial is K[x^2-(α+β)x+αβ]

=> K[x^2-{(4/3)+(9/2)}x+(4/3)(9/2)]

=>K[x^2-{(8+27)/6}x+(6)]

=> K[x^2-(35/6)x+6]

=> K[6x^2-35x+36]/6

If K = 6 then The polynomial = 6x^2-35x+36

Answer:-

The required quadratic polynomial whose zeroes are alpha²/beta and beta²/ alpha is 6x^2-35x+36

Used formulae:-

• the standard quadratic Polynomial ax^2+bx+c

• If α and β are the zeroes them the Quadratic Polynomial is K[x^2-(α+β)x+αβ]