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1)If all the roots (zeros) of the polynomial f(x)=x^5+ax^4+bx^3+cx^2+dx-420 are integers larger than 1, then f(4)=? 2)Consider the two functions f(x) =x^2+2bx+1 and g(x)=2a(x+b), where the variable x and the constants a and b are real numbers. Each such pair of the constants a and b may be considered as a point (a,b) in an ab - plane.Let S be the set of such points (a,b) for which the graphs of y=f(x) and y=g(x) do not intersect (in the xy- plane). The area of S =? 3)if the equation 2x^2 +4xy + 7y^2-12x-2y+t=0 where “t’ is a parameter has exactly one real solution of the form (x,y).Then the sum of (x+y) =?

1)If all the roots (zeros) of the polynomial f(x)=x^5+ax^4+bx^3+cx^2+dx-420 are integers larger than 1, then f(4)=?
 
2)Consider the two functions f(x) =x^2+2bx+1 and g(x)=2a(x+b), where the variable x and the constants a and b  are real numbers. Each such pair of the constants a and b may be considered as a point (a,b) in an ab - plane.Let S be the set of such points (a,b) for which the graphs of y=f(x) and y=g(x) do not intersect (in the xy- plane). The area of S =? 
 
3)if the equation 2x^2 +4xy + 7y^2-12x-2y+t=0 where “t’ is a parameter has exactly one real solution of the form (x,y).Then the sum of (x+y) =?

Grade:11

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
Hello student,
Please find the answer to your question below
f must have 5 (not necessarily distinct) roots d1, d2, . . . d5.
f factors as
(x – d1)(x – d2)(x – d3)(x – d4)(x – d5).
The product d1⋅ d2⋅ d3⋅ d4⋅ d5 must be equal to420,
which factors as 2,2,3,5,7.
All of the roots are integers larger than 1, so they must
be 2, 2, 3, 5, and 7.
So f(x) = (x – 2)2(x – 3) (x – 5) (x – 7).................(1)
Putting in x = 4 in equation (1) we get x=12.

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