Question

find the quadratic equation whose zeros are square of the zeros of the polynomial x² - x - 1.

Answer 1

Now

let the roots of this polynomial be denoted by m and n.

we know that the general form of a quadratic polynomial is

x2−sx+p.....(2)x2−sx+p.....(2)

where s=sum of roots and p=product of roots

Now

comparing the general form with (1) we get

s=1 and p=-1

this implies s=sum of roots =1=m+n=1=m+n……(3)……(3)

and p=product of roots =−1=mn…….(4)=−1=mn…….(4)

Now

let the required polynomial be

x2−Sx+P.......(5)x2−Sx+P.......(5)

and let k and t denote its roots

Now (5) can also be written as

x2−(k+t)x+ktx2−(k+t)x+kt.......(6).......(6)

according to the question

k=m2andt=n2k=m2andt=n2 (zeroes of the required polynomial are the square of the zeros of the given polynomial)

putting these in (6) we get

x2−(m2+n2)x+(mn)2x2−(m2+n2)x+(mn)2

Now above equation can also be written as

x2−((m+n)2–2mn)x+(mn)2x2−((m+n)2–2mn)x+(mn)2……….(7)……….(7)

substituting the values from (3) and (4) in (7) we get

x2−((1)2–2(−1))x+(−1)2x2−((1)2–2(−1))x+(−1)2

=x2−(3)x+1=x2−(3)x+1

=x2–3x+1=x2–3x+1

which is the required polynomial .

hope it helps

:-)

Answer 2

If roots of the equation p(x)^2+qx+r=0 are a,b. The equation with roots (a)^2,(b)^2 is p((x)^2)^1/2+q(x)^1/2+r=0