a) Show that f has no rational roots. b) Determine the remainder of the division of f by g. c) Show that: s(cos) CoS 9. -
Consider polynomials $f, g \in Q[X]$,
$$
\begin{array}{l}
f=8 X^{3}-6 X-1 \\
g=4 X^{3}-3 X
\end{array}
$$
and the set:
$$
M=\left\{h\left(\cos \frac{\pi}{9}\right) \mid h \in Q[X]\right\}
$$
a) Show that f has no rational roots.
b) Determine the remainder of the division of f by g.
c) Show that:
$$
f\left(\cos \frac{\pi}{9}\right)=0
$$
d) Show that:
$$
\cos \frac{\pi}{9} \in \mathrm{R}-Q
$$
e) If $h \in Q[X]$ and
$$
h\left(\cos \frac{\pi}{9}\right)=0
$$
show that h is divided by f.
This problem with many subpoints all are for one problem.
Please check the attached pictures for clarity, and please do not answer the question if you are not sure that your answer is corect because you will make me lose a problem that I have already paid for, thank you for your understanding.
![Consider polynomials f, g € Q[X],
f = 8X3 – 6X – 1
g = 4X³ – 3X
and the set:
M = {h(cos) |he Q[X}}
T
a) Show that f has no rational roots.
b) Determine the remainder of the division of f by g.
c) Show that:
Cos –
d) Show that:
ER – Q
COS
e) If h E Q[X] and
A (cos 5)
T
h( cos
= 0
show that h is divided by f.
f) Show that M is a three-dimensional vector space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde54a8a1-129b-452f-b5d1-f7f157f4c8c4%2F8d8396d7-0fe6-49fb-b794-72af1c5c443f%2F3z7itgb_processed.jpeg&w=3840&q=75)
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