Now we want to find the zeros of the polynomial function.
g(x)=6x(x-1)(x+6)
To do so, we need to find the values of x for which g(x)=0.
g(x)=0 ⇔ 6x(x-1)(x+6)=0
We will use the Zero Product Property to solve the equation.
We found that the zeros of the function are 0, 1 and - 6.
Additional Points and End Bahaviour
To draw the graph of the function, we will find some additional points and consider the end behaviour. Let's use a table to find additional points.
x
6x^3+30x^2-36x
g(x)=6x^3+30x^2-36x
- 5
6( - 5)^3+30( - 5)^2-36( - 5)
180
- 3
6( - 3)^3+30( - 3)^2-36( - 3)
216
- 1
6( - 1)^3+30( - 1)^2-36( - 1)
60
0.5
6( 0.5)^3+30( 0.5)^2-36( 0.5)
- 9.75
2
6( 2)^3+30( 2)^2-36( 2)
96
The points ( - 5, 180), ( - 3, 216), ( - 1, 60), ( 0.5, - 9.75) and ( 2, 96) are on the graph of the function. Finally, as we already have the standard form of our function we can determine the leading coefficient and degree of the polynomial function.
g(x)= 6x^3+30x^2-36x
We can see that the leading coefficient is 6, which is a positive number. Also, the degree is 3, which is an odd number. Therefore, the end behavior is down and up.
Graph
Now, let's draw the graph through our points, considering the end behaviour.
