SOLVING POLYNOMIAL INEQUALITIES USING ALGEBRA

There are 2 ways to solve a polynomial inequality using algebra. Interval Method, and Case Method.

 

Interval Method

 

The interval method has 3 steps:

 

 

 

Example

 

-2x^3 + x^2 + 13x + 6 < 0

 

Eliminate the (-) by dividing both sides by (-1):

2x^3 + x^2 + 13x + 6 > 0

 

Use (x +2) as the first factor. Found by substituting factors of (-6) into the function to check which value gives the remainder of zero because that means that that is a factor. After doing synthetic division the result is:

(x+2)(2x^2 -5x -3) > 0

 

After fully factoring:

(x + 2)(2x + 1)(x – 3) > 0

 

Now create the table:


 

X < -2

-2 < X < (-1/2)

-(1/2) < X < 3

X > 3

 

 

 

 

 

Test Value:

-3

-1

2

4

 

 

 

 

 

X + 2

(-)

(+)

(+)

(+)

2X + 1

(-)

(-)

(+)

(+)

X - 3

(-)

(-)

(-)

(+)

 

 

 

 

 

Sum:

(-)

(+)

(-)

(+)

 

In the table above there are a lot of things happening:

 

 

 

 

 

 

The sum is the indication to whether the function will be negative of positive within a certain interval. Since the required information is when the function will be greater than 0, it is easy to see that the function is positive when x is between (-2, -(1/2)) and (3, ∞).

 

Therefore the solution is: F(x) > 0 when XE(-2,(-1/2))U(3, ∞)

 

 

If you have any questions, leave them in the comments below.