solve the polynomial inequality and graph the solutionset ona real number line, express the solutionset in interval notation x^3-4x^2-16x+64<0
To solve the inequality, we need to find the roots of the associated polynomial `f(x)=x^3-4x^2-16x+64.`
Factoring the polynomial, we get:
This means that the zeros of `f(x)` are x=-4 and x=4.
The real number line is now divided into three regions: `(-infty,-4)` , `(-4,4)` and `(4,infty)` .
For the left-most region, we take a point such as x=-5 and test the polynomial to get `f(-5)=(-9)^2(-5+4)=-81<0`
so this region is a solution of the inequality.
The centre region can be tested using a point, such as x=0 to get `f(0)=64>0` so this region is not a solution to the inequality.
The right-most region can also be tested using a point such as x=5, to get `f(5)=(5-4)^2(5+4)=9>0`
so this region is also not a solution to the inequality.
The solution to the inequality is `(-infty,-4)` . A graph of the number line with the solution is