We can see in Equation (I) that x=0 is a solution for the given equation. To find other solutions, we will solve Equation (II).
x^2+x-6=0
Note that the above equation is a quadratic equation. Thus, we can solve it using the Quadratic Formula.
ax^2+bx+c=0 ⇔ x=- b±sqrt(b^2-4ac)/2a
Let's identify a, b, and c.
x^2+x-6=0 ⇔ 1x^2+ 1x+( -6)=0
We see that a= 1, b= 1, and c= -6. Let's substitute these values into the formula, and solve for x.
The solutions for this equation are x= -1± 52. Let's separate them by using the positive and negative signs.
x=-1± 5/2
x=-1+5/2
x=-1-5/2
x=4/2
x=-6/2
x=2
x=-3
We found that the zeros of f(x)=x^3+x^2-6x are x=- 3, x=2, and x=0.
Graph
To draw the graph of the function, we will find some additional points and consider the end behavior. Let's use a table to find additional points.
x
x^3+x^2-6x
f(x)=x^3+x^2-6x
- 2
( - 2)^3+( - 2)^2-6( - 2)
8
1
1^3+( 1)^2-6( 1)
- 4
3
3^3+( 3)^2-6( 3)
18
The points ( - 2, 8), ( 1, - 4), and ( 3, 18) are on the graph of the function. We can also see that the leading coefficient is 1, which is a positive number. Also, the degree is 3, which is an odd number. Therefore, the end behavior is down and up. Now, let's draw the graph!
