How do you solve the polynomial inequality and state the answer in interval notation given $x^4<=16+4x-x^3$ ?

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Answer 1 · 2017-11-25T13:07:10

The solution is $[-2,2]$

Let us use a simplified version of 'sign chart' method.

$x^4<=16+4x-x^3$ can be written as $x^4+x^3-4x-16<=0$

As $2$ and $-2$ are the zeros

$x^4+x^3-4x-16=(x-2)(x+2)(x^2+x+4)$

Observe that $x^2+x+4$ is always positive (note that $x^2+x+4=(x+1/2)^2+15/4$ - sum of two positive numbers) and hence

whether $x^4+x^3-4x-16<=0$ depends on three intervals

$x<=-2$ , $-2<=x<=2$ and $x>=2$

Let us pick up three values each in one of the range,

say $x=-4$ , $x=0$ and $x=4$

as only $x=0$ satisfies the inequality,

solution is $-2<=x<=2$ and in interval notation $[-2,2]$

graph{x^4+x^3-4x-16 [-10, 10, -5, 5]}

How do you solve the polynomial inequality and state the answer in interval notation given x^4<=16+4x-x^3x^4<=16+4x-x^3?