Solve Polynomial Inequality...2

[nycmathguy]

Section 2.7
Question 36

x^4(x - 3) ≤ 0

Set each factor to zero and solve for x.

(x^4)^(1/4) = (0)^(1/4)

x = 0

x - 3 = 0

x = 3

The key numbers are x = 0 and x = 3.

Our real number line looks like this:

<--------------0------------3------------->

We now select a number from each interval to evaluate in the given polynomial inequality.

Let x = -1

(-1)^4(-1 - 3) ≤ 0

-4 ≤ 0...true statement.

Let x = 0

(0)^4(0 - 3) ≤ 0

0 ≤ 0...true statement.

Let x = 1

(1)^4(1 - 3) ≤ 0

-2 ≤ 0...true statement.

Let x = 3

(3)^4(3 - 3) ≤ 0

0 ≤ 0...true statement.

Let x = 4

(4)^4(4 - 3) ≤ 0

256 ≤ 0...true statement.

As we can see, the polynomial inequality is satisfied in the interval (-infinity, 3].

See attachment for graph of the solution set.

 

[MathLover1]

you needed to do just this:
x^4(x - 3) ≤ 0
do each factor
x^4≤ 0=> x≤ 0
(x - 3)≤ 0
x ≤ 3

your solution is correct and number line is also correct

 

[nycmathguy]

you needed to do just this:
x^4(x - 3) ≤ 0
do each factor
x^4≤ 0=> x≤ 0
(x - 3)≤ 0
x ≤ 3

your solution is correct and number line is also correct

I got it right but your way is far easier.

 

[MathLover1]

you say: (x^4)^(1/4) = (0)^(1/4) how did you get it?

 

[nycmathguy]

you say: (x^4)^(1/4) = (0)^(1/4) how did you get it?

I see two factors: x^4 and (x - 3). By setting each factor to and solving for x, I must take the fourth root on each side for x^4 = 0. In place of taking the fourth root of x^4, I simply raised each side to the (1/4) power.

(x^4)^(1/4) = x.

 

[MathLover1]

ok, but there is no need for that
if x^4=0, then x could be only 0 to satisfy equality

 

[nycmathguy]

ok, but there is no need for that
if x^4=0, then x could be only 0 to satisfy equality

You are right but I like to be mathematical in most of my threads.