Section 3-2: Polynomial Inequalities, One Variable
 
 
 
We have two methods to solve polynomial inequalities:
 
   1)  Find the zeros and use a sign analysis of P(x)
 
   2)   Use the graphing calculator and analyze the graph of P(x)
 

 
Method One!
 
1)  x2 + x - 6 > 0
First, factor the polynomial.
(x + 3)(x - 2) > 0
Identify the zeros!
x = -3, x = 2
Graph the zeros, use open dots (strickly greater than)
Now do a sign analysis by picking a point in each of the three regions divided by the zeros of the function.
 
Chosen x value Sign of the function
-4 (-4 + 3)(-4 - 2),   +
0 (0 + 3)(0 - 2),     -
3 (3 + 3)(3 - 2),    +
 
The sections we want are the positive sections because the function is greater than zero!
Therefore, the answer is:
x < -3 or x > 2

2)  (x2 - 4)(x - 1)2 < 0
Factor the polynomial to find the zeros
(x - 2)(x + 2)(x - 1)2 < 0
The zeros are: x = 2, x = -2, x = 1 (double root)
Do a sign analysis for the four sections divided by the three zeros
 
Chosen x value Sign of function
-3 (-3 - 2)(-3 + 2)(-3 - 1)2,   +
0 (0 - 2)(0 + 2)(0 - 1)2,      -
1.5 (1.5 - 2)(1.5 + 2)(1.5 - 1)2, -
3 (3 - 2)(3 + 2)(3 - 1)2,     +
 We want the sections that are negative because the polynomial is less than 0.
Therefore, the answer is:
-2 < x < 2

 
3)  (x - 3)(x + 4)/(x - 5) > 0
This is already factored.  Find the zeros of the numerator and denominator but keep in mind the bottom can never equal zero, so the zeros of the denominator will always have open dots.
The zeros are: x = 3, x = -4 and x = 5
Now do your sign analysis for the four sections seperated by the three zeros.
 
Chosen x value sign function
-5 (-5 - 3)(-5 + 4)/(-5 - 5),   -
0 (0 - 3)(0 + 4)/(0 - 5),       +
4 (4 - 3)(4 + 4)/(4 - 5),       -
6 (6 - 3)(6 + 4)/(6 - 5),       +
The sections that we want are positive because the polynomial is greater than zero.  Therefore, the solutions are:
-4 < x < 3 or x > 5
 

 
Method two!
 
This method is accomplished by using your graphing calculator!!  Graph the function and identify the zeros.  Then, any part above the x-axis is the positive area (> 0) and any part below the x-axis represents the negative area(< 0).
 
1)  2x3 + x2 - 5x - 2 < 0
Use your graphing feature on the TI-82 to graph this function.  Only type in upto the < symbol.  This is the graph:
Use the calc section choice 2 to find each of the zeros.  The zeros (rounded to tenths place) are:
x = -1.6, x = -.4, and x = 1.5
We want the area that is less than zero, or the area below the x axis.  Thus, the solution is:
x < -1.6 or -.4 < x < 1.5
This is a good method to use when you can't factor the polynomial!!