Prove that the polynomial $x^6+x^4-5x^2+1$ has at least four real roots.

Question

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.

Answer 1

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 $(1,2)$.

Answer 2

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

Answer 3

Consider $y=x^2$, and the polynomial $f(y)=y^3+y^2-5y+1$.

$f(0)=1>0$, $f(1)=-2<0$, $f(100)>0$ since $f(y)\to \infty $ as $y\to \infty$. Hence $f(y)$ has at least two positive roots.

Answer 4

Hint Prove that $g(y)=y^3+y^2-5y+1$ has at least two positive real roots.

This follows immediately from $g(0)>0, g(1) <0$.