{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 461hw6 - why f a,b,c,d is a real number(Hint The...

This preview shows page 1. Sign up to view the full content.

Math 461 F Spring 2011 Homework 6 Drew Armstrong Reading. Section 6.4, 6.5 Book Problems. 6.4: 1, 4, 8. Additional Problems. A.1. Euler showed that every real polynomial of the form x 4 + αx 2 + βx + γ factors into two real quadratics. Use his result to prove that every real polynomial of the form ax 4 + bx 3 + cx 2 + dx + e factors into two real quadratics. (Hint: Use a change of variables to turn your polynomial into Euler’s polynomial.) A.2. Suppose that a given real quartic equation has roots a,b,c,d in some ﬁeld E R . (Today we know that these roots must be complex, but in times past their nature was mysterious.) Now let f ( a,b,c,d ) be some function of a,b,c,d that is invariant under any permutation of the roots. Explain
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: why f ( a,b,c,d ) is a real number. (Hint: The Fundamental Theorem of Symmetric Functions.) A.3. Let a,b,c,d be the roots of some real quartic equation with no x 3 term (i.e. we have a + b + c + d = 0.) Let p = a + b , q = a + c , and r = a + d , so that-p = c + d ,-q = b + d , and-r = b + c . Prove that pqr is a real number, and hence-p 2 q 2 r 2 is a negative real number. (Hint: Show that pqr is invariant under permuting a ↔ b , or a ↔ c , or a ↔ d . Hence it’s invariant under any permutation of a,b,c,d .)...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online