Question

If zeros ff(x) =x cube -3x square +x+1 are a-b, a, a+b find a and b

Answer 1

We are given the polynomial

f(x)=x^3-3x^2+x+1

Zeroes of this polynomial=alpha=a+b , beta =a , gama=a-b

Zero of any cubic polynomial includes three cases:-

Case (i) =alpha +beta +gama= -Coefficient of x^2/coefficient of x^3

a+b+a+a-b=-(-3)/1

3a=3

a=1

Case ii=alpha×beta+beta×gama+gama×alpha=coefficient of x/coefficient of x^3

=(a+b)(a)+(a)(a-b)+(a-b)(a+b)=1/1

a^2+ab+a^2-ab+a^2-b^2=1

3a^2-b^2=1

now putting value of a in above equation

3-b^2=1

-b^2=-2

b=root 2

Case iii-alpha× beta×gama=-constant term/coefficient of x^3

a+b×a×a-b=-1/1

a^2-b^2×a=-1

a^3-ab^2=-1

1-1×(root 2)^2=-1

1-2=-1

-1=-1

Hence

a=1 and b=root 2