Prove that the polynomial x 6 + x 4 − 5 x 2 + 1 has at least four real roots.Talking analysis here, using the definition of continuity, intermediate value theorem, and extreme value theorem.

Question

Prove that the polynomial       x    6    +      x    4    −  5      x    2    +  1 has at least four real roots.Talking analysis here, using the definition of continuity, intermediate value theorem, and extreme value theorem.

neglegir86 · Accepted Answer

Giving your polynomial the name   f,  f  (  −  2  ) is positive.  f  (  −  1  ) is negative.  f  (  0  ) is positive.  f  (  1  ) is negative.  f  (  2  ) is positive.So the intermediate value theorem says there must be a root in each of   (  −  2  ,  −  1  ),   (  −  1  ,  0  ),   (  0  ,  1  ), and   (  1  ,  2  )

Flambergru · Accepted Answer

Hint: the function is even, so you just need to show it has at least two positive real roots.

Prove that the polynomial x^6+x^4−5x^2+1 has at least four real roots. Talking analysis here, using the definition of continuity, intermediate value theorem, and extreme value theorem.

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