Polynomials

Question 1
 
A
1
B
2
C
3
D
4
       Engineering-Mathematics       Polynomials       Gate-2000
Question 1 Explanation: 
P(x) should be like

Minimum degree of P(x) = 4
Maximum degree of P(x) = 5
Question 2
A polynomial p(x) is such that p(0) = 5, p(1) = 4, p(2) = 9 and p(3) = 20. The minimum degree it can have is
A
1
B
2
C
3
D
4
       Engineering-Mathematics       Polynomials       Gate-1997
Question 2 Explanation: 
Lets take p(x) = ax + b
p(0) = 5 ⇒ b = 5
p(1) = 4 ⇒ a+b = 4 ⇒ a = -1
p(2) = 9 ⇒ 40+b = 9 ⇒ -4+5 = 9, which is false.
So degree 1 is not possible.
Lets take p(x) = ax2 + bx +c
p(0) = 5 ⇒ c = 5
p(1) = 4 ⇒ a+b+c = 4 ⇒ a+b = -1 -----(1)
p(2) = 9 ⇒ 4a+2b+c = 9 ⇒ 2a+b = 2 -----(2)
(2) - (1)
⇒ a = 3, b = -1-1 = -4
p(3) = 20 ⇒ 9a+3b+c = 20
⇒ 27-12+5 = 20
⇒ 20 = 20, True
Hence, minimum degree it can have.
Question 3
 
A
All complex roots
B
At least one real root
C
Four pairs of imaginary roots
D
None of the above
       Engineering-Mathematics       Polynomials       GATE-1987
Question 3 Explanation: 
Since, the polynomial has highest degree 7. So there are 7 roots possible for it.
Now suppose if an imaginary number a+bi is also root of this polynomial. That means there must be even number of complex root possible because, they occur in pair.
A) All complex root.
This is not possible. The polynomial has 7 roots and as I mention a polynomial should have been number of complex root and 7 is not even. So this option is wrong.
B) At last one real root.
This is possible. Since polynomial has 7 roots and only even number of complex root is possible, that means this polynomial has max 6 complex roots and hence minimum one real root. So, this option is correct.
C) Four pairs of imaginary roots.
4 pair means 8 complex root. But this polynomial can have atmost 7 roots. So, this option is also wrong.
Question 4
If f(xi) ⋅ f(xi+1) < 0 then
A
There must be a root of f(x) between xi and xi+1.
B
There need not be a root of f(x) between xi and xi+1.
C
There fourth derivative of f(x) with respect to x vanishes at xi.
D
The fourth derivative of f(x) with respect to x vanishes at xi+1.
       Engineering-Mathematics       Polynomials       GATE-1987
Question 4 Explanation: 
As f(xi) ⋅ f(xi+1) < 0
means one of them is positive and one of them is negative, as their multiplication is negative.
So, when you draw the graph for f(x) where xi ≤ x ≤ xi+1.
Definitely f(x) will cut the x-axis. So there will definitely be a root of f(x) between xi and xi+1.
There are 4 questions to complete.