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]}