Use the Intermediate Value Theorem and Rolle's Theorem to show the that the polynomial p ( x ) = x 5 + x 3 + 7 x − 2has a unique real root.Can someone please give some hints on how to do this question.

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Use the Intermediate Value Theorem and Rolle&#039;s Theorem to show the that the polynomial  p  (  x  )  =      x          5        +      x          3        +  7  x  −  2has a unique real root.Can someone please give some hints on how to do this question.

persstemc1 · Accepted Answer

The Intermediate Value Theorem establishes existence: there is at least one real root.Notice that   p  (  0  )  =  −  2  &amp;lt;  0 and   p  (  1  )  =  7  &amp;gt;  0. Since p is continuous, the I.V.T. guarantees a number c such that   p  (  c  )  =  0. (In fact, we know that   0  &amp;lt;  c  &amp;lt;  1.)Rolle&#039;s Theorem establishes uniqueness: there is at most one real root. Why? Suppose that there were two roots   a  ,  b  ∈      R  . Since p is differentiable, Rolle&#039;s Theorem guarantees a number   c  ∈  (  a  ,  b  ) where       p    ′    (  c  )  =  0. What&#039;s wrong with that? The derivative      p    ′    (  x  )  =  5      x    4    +  3      x    2    +  7  &amp;gt;  0for all   x  ∈      R  . Why? It&#039;s quadratic in x2 and its discriminant   (  3      )    2    −  4  (  5  )  (  7  )  &amp;lt;  0.

Use the Intermediate Value Theorem and Rolle's Theorem to show the that the polynomial p (

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