How would you show that the polynomial $\displaystyle p(x) = x^4 + 7 x^3-9$ has at least two real roots?

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- Mar 1st 2010, 04:24 PMCrazyCat87Polynomial Roots
How would you show that the polynomial $\displaystyle p(x) = x^4 + 7 x^3-9$ has at least two real roots?

- Mar 1st 2010, 04:36 PMBlack
p(-8) > 0

p(0) < 0

p(2) > 0

Since any polynomial is continuous, it follows from the intermediate value theorem. - Mar 1st 2010, 06:35 PMCrazyCat87
- Mar 1st 2010, 06:51 PMBlack
By the IVT, since p(0) < 0 < p(-8) and p(0) < 0 < p(2), there exist real numbers c in [-8,0] and d in [0,2] such that p(c)= p(d) = 0.

- Mar 1st 2010, 09:10 PMCaptainBlack