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Grade: 11
        
The coefficients p,q of the expression x4 + px3 + (q+1)x2 + px + q are drawn from the set {1,2,3,4,5,6}. Find the  number of ways in which it can be done so that the expression is positive for all real x.
 
 
11 months ago

Answers : (1)

Aditya Gupta
1673 Points
							
The expression can be factored as
(x^2+1)(x^2+px+q)
So for this to be always positive, x^2+px+q needs to be always positive. That means the discriminant must be less than zero.
Or p^2 less than 4q.
When p is 1, q can take all 6 values.
When p is 2, q can take 5 values.
When p is 3, q can take 4 values.
When p is 4, q can take 2 values.
When p is 5 or 6, q can not take any value.
So total number of ways to select p and q is:
6+5+4+2
=17
10 months ago
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