Question

All roots of x^4 - 12x^3 + ax^2 + bx + 81 = 0 are non-negative. the ordered pair (a,b) can be

Answer 1

Let c,d,e,f be the roots of the above equation.

Observe that c+d+e+f=12 and cdef=81

and (c+d+e+f)/4≥(cdef)^{1/4} (By AM-GM inequality)

as (c+d+e+f)/4=(cdef)^{1/4}, so, c=d=e=f=12/4=3

So, a=6(3×3)=6×9=54 and b=-4(3×3×3)=-108

Answer 2

Given: roots of x^4 - 12x^3 + ax^2 + bx + 81 = 0 are non-negative

To find: the ordered pair (a,b) can be

Solution:

Let c, d, e, f be the roots of the above equation.

By Sum of roots:

• c + d + e + f = 12

By Product of roots:

• c x d x e x f = 81

Now, by applying Athematic and Geometric Mean Inequality formula,

• (c + d + e + f) / 4 ≥ ( c x d x e x f ) ^ {1/4}          

As,

• (c + d + e + f) / 4 = ( c x d x e x f) ^ {1/4}

Therefore,

• c = d = e = f = 12 / 4 = 3

So, a = 6 x (3×3) = 6 × 9 = 54

and

b = - 4 ( 3 × 3 × 3 ) = -108

Ordered pair (a, b) are (54, -108)