Consider the polynomial: f = 2X² +aX³ + 3X² +bX+c= R[X] with roots X₁, X2, X3, X4 E C. a) Calculate: Σ (x₁ - xj)² 1
Consider the polynomial:
$$
f=2 X^{4}+a X^{3}+3 X^{2}+b X+c \in \mathbf{R}[X]
$$
with roots $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{X}_{3}, \mathrm{X}_{4} \in \mathbb{C}$.
a) Calculate:
$$
\sum_{1 \leq i<j \leq 4}\left(x_{i}-x_{j}\right)^{2}
$$and then prove that for | a | <4 the polynomial has at most two real roots.
b) For | a | = 4 determine b and c so that f has all roots real.
c) For | a |> 4 prove that there is b and c in $\mathbb{R}$ such that f has all roots real.
Please check the attached picture for details.
I need complete solution with explanation please.
All three subpoints are subpoints for the same problem and they are conected between them, but if it is not possible to solve them all please solve what you can, thank you in advance.
![Consider the polynomial:
f = 2x4 + aX3+ 3X? + bX + c € R[X]
with roots x1, X2, X3, X4 E C.
a) Calculate:
|
1<i<j<4
and then prove that for | a | <4 the polynomial has at
most two real roots.
b) For | a| = 4 determine b and c so that f has all roots
real.
c) For | a |> 4 prove that there is b and c in R such that
f has all roots real.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde54a8a1-129b-452f-b5d1-f7f157f4c8c4%2F039e81a7-b6dc-4d5b-bd34-87c892a59a12%2Fxbcks8_processed.jpeg&w=3840&q=75)
Step by step
Solved in 3 steps
