Find the point of intersection of the given plane (y = −17) and the line that is perpendicular to the given plane and passes through (4, 8, 3).

The point of intersection is

Step-by-step explanation:

In order to find the intersection point we will need to find the vector equation of the line.

The plane equation is

Now to find the line that is perpendicular to this plane, we can do the following reasoning :

The plane and the line will be perpendicular between each other if and only if the direction vector of the line is a scalar multiple from the normal vector of the plane.

The plane equation is . Then any vector with the following form :

With ∈ IR , will be perpendicular to the plane.

Let's choose to simplify the calculations.

The normal vector of the plane is

Now the direction vector of the line must be a scalar multiple of this vector ⇒    , ∈ IR

Let's also choose in order to simplify the calculations.

The vectorial equation of the line is :

     with ∈ IR

Where is the direction vector and is any point where the line passes through.

Using the data from the question we complete our equation with :

    with ∈ IR

Now let's find the intersection point between the plane and the line

This intersection point must belong to the line and also to the plane.

In order to belong to the plane its second component must be

Knowing this and using the vector equation of the line we obtain this equation :

⇒

Using in the vector equation of the line we find that the point is

This point belongs to the plane (because its second component is ) and also to the line (because it can be obtained using the vector line equation with ).

f(x) = x^3 -2x^2 -3x

x( x^2 -2x-3)

x( x^2 +x -3x -3)

x( x(x+1) -3(x+1))

x( x-3)( x+1)

so zeroes are 0, 3,-1

its (2) option