Let f ( x ) be a monic polynomial of odd degree. Prove that ∃ A ∈ R s.t. f ( A ) &lt; 0 and there exists B ∈ R such that f ( B ) &gt; 0.Deduce that every polynomial of odd degree has a real root.There are questions that answer the final part, but they do not do so by proving the first part. I am fairly sure that this involves the intermediate value theorem, but not sure how to implement it in this case.

Question

Let   f  (  x  ) be a monic polynomial of odd degree. Prove that   ∃  A  ∈      R   s.t.   f  (  A  )  &amp;lt;  0 and there exists   B  ∈      R   such that   f  (  B  )  &amp;gt;  0.Deduce that every polynomial of odd degree has a real root.There are questions that answer the final part, but they do not do so by proving the first part. I am fairly sure that this involves the intermediate value theorem, but not sure how to implement it in this case.

RormFrure6h1 · Accepted Answer

For the final part. If you have   f  (  A  )  &amp;lt;  0 and   f  (  B  )  &amp;gt;  0 then by the IVT every value in [f(A),f(B)] is attained by f(x) for some x between A and B, and this includes 0.To show the existence of the A and B show that for x large one has that the sign of f(x) is the sign of the leading coefficient. And, if the degree is odd for small x one has that the sign of f(x) is the opposite sign of the leading coefficient.

Let f <mo stretchy="false">( x <mo stretchy="false">) be a monic polynomial of odd de

Answered question

Answer & Explanation

New Questions in Calculus 2