Question

If α and β are the zeroes of the polynomial f(x) = x² + px + q, form a polynomial whose zeroes are (α + β)² and (α − β)².

Answer 1

If α and β are the zeros of the quadratic polynomial f(x) = x² + px + q

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

`=(-p)/1`

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

`=q/1`

= q

Let S and P denote respectively the sums and product of the zeros of the polynomial whose zeros are (α + β)² and (α − β)²

S = (α + β)² + (α − β)²

`S=alpha^2+beta^2+2alphabeta+alpha^2+beta^2-2alphabeta`

`S=2[alpha^2+beta^2]`

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

`S=2(p^2-2xxq)`

`S=2(p^2-2q)`

 

`P=(alpha+beta)^2(alpha-beta)^2`

`P=(alpha^2+beta^2+2alphabeta)(alpha^2+beta^2-2alphabeta)`

`P=((alpha+beta)^2-2alphabeta+2alphabeta)((alpha+beta)^2-2alphabeta-2alphabeta)`

`P=(p)^2((p)^2-4xxq)`

`P=p^2(p^2-4q)`

The required polynomial of f(x) = k(kx² - Sx + P) is given by

f(x) = k{x² - 2(p² - 2q)x + p²(p² - 4q)}

 

f(x) = k{x² - 2(p² - 2q)x + p²(p² - 4q)}, where k is any non-zero real number.