# Polynomial Roots

• March 1st 2010, 04:24 PM
CrazyCat87
Polynomial Roots
How would you show that the polynomial $p(x) = x^4 + 7 x^3-9$ has at least two real roots?
• March 1st 2010, 04:36 PM
Black
p(-8) > 0
p(0) < 0
p(2) > 0

Since any polynomial is continuous, it follows from the intermediate value theorem.
• March 1st 2010, 06:35 PM
CrazyCat87
Quote:

Originally Posted by Black
p(-8) > 0
p(0) < 0
p(2) > 0

Since any polynomial is continuous, it follows from the intermediate value theorem.

How does that prove there's at least 2 roots??
• March 1st 2010, 06:51 PM
Black
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.
• March 1st 2010, 09:10 PM
CaptainBlack
Quote:

Originally Posted by CrazyCat87
How would you show that the polynomial $p(x) = x^4 + 7 x^3-9$ has at least two real roots?

Descartes rule of signs show that this has one positive root, it also shows that this has one negative root. Hence it has at least two real roots.

CB