IIT JEE , AIEEE Forum - Mathematics , Algebra

man111

(2) if f(x)=x⁴+17x³+80x²+203x+125=0

than find a min polynomial g(x)

such that f(3+-3) = g(3+-3)

and f(5+-5) = g(5+-5).

means f(3+3) = g(3+3) and f(3-3) = g(3-3) same for second f(5+5) = g(5+5) and f(5-5) = g(5-5)

hsbhatt

Consider the polynomial h(x) = f(x) - g(x).

we are given that the zeros of h(x) are a = 3+

3, b =3-

3, c= 5+

5, and d= 5-

5

Hence, h(x) = P(x) (x-a)(x-b)(x-c)(x-d).

The least degree polynomial is (x-a)(x-b)(x-c)(x-d) = (x²-(a+b)x+ab) (x²-(c+d)x+cd)

= (x^2-6x+6) (x²-10x+20)

= x⁴-16x³+86x²-180x+120

Hence, g(x) = f(x) - h(x) = 33x³-6x²+383x+5 is one instance of a poynomial satsifying the given conditions.

sboosy

excelllent bhattji