Consider the polynomial: f = 2X² +aX³ + 3X² +bX+cER[X] with roots X₁, X2, X3, X4 E C. a) Calculate: Σ (; - z;)2 1 4 prove that there is b and c in R such that f has all roots real.
Consider the polynomial: f = 2X² +aX³ + 3X² +bX+cER[X] with roots X₁, X2, X3, X4 E C. a) Calculate: Σ (; - z;)2 1 4 prove that there is b and c in R such that f has all roots real.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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.
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I need complete solution with explanation please.
![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%2F9002d13e-5daf-4c4e-a9fe-36cbab25cd96%2Fgxia9zp_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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.
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