z^3 + 7z^2 + 10z > 0

Question

Solving polynomial inequalities- find and graph the solution set

Answer 1

 

Let A=z(z²+7z+10)=z(z+5)(z+2)

TEST z	z	z+5	z+2	A
-6 (z<-5)	<0 (-)	<0 (-)	<0 (-)	<0
-3 (-5<z<-2)	<0 (-)	>0 (+)	<0 (-)	>0
-1 (-2<z<0)	<0 (-)	>0 (+)	>0 (+)	<0
1 (z>0)	>0 (+)	>0 (+)	>0 (+)	>0

The SIGN TABLE above shows the sign of A for all values of z.

Solutions: -5<z<-2 and z>0. The graph of the function with sign markings is shown below:

 

Answer 2

z^3 + 7z^2 + 10z > 0

z(z^2 + 7z + 10) > 0

z(z + 2)(z + 5) > 0

Solution set is given by,

all 3 terms +ve

1 term +v e and 2 terms -ve

This can be modelled as

+   +   +   > 0  -->   z > 0   (z + 2) > 0   (z + 5) > 0 ->   z > 0

+   -   -   > 0   -->   z > 0   (z + 2) < 0   (z + 5) < 0 ->    no solution

-   +   -   > 0   -->   z < 0   (z + 2) > 0   (z + 5) < 0 ->    no solution

-   -   +   > 0   -->   z < 0   (z + 2) < 0   (z + 5) > 0 ->    -5 < z < -2

There are thus two solution sets: z > 0 and -5 < z < -2