Draw the graph of x^3 + x^2 - 6x

This is a cubic curve because the highest power to which x is being raised is 3. Cubic curves with addition x^2 and x terms often have a distinctive shape - sort of like a capital 'N' drawn using curves.

So, we stand a good chance of being able to draw this curve if we find out where it passes through the x-axis. To do this, we need to factorise its equation.

We can immediately see that we can factorise out an x term all by itself, as every term in the curve's equation has x as a factor. Doing this, we can see that the curves equation is equal to (x)(x^2 + x - 6). We now need to factorise the second part of this equation, namely, x^2 + x - 6.

Remember that when we factorise a quadratic expression in the form ax^2 + bx + c we are wanting to end up with two expressions which fulfil certain conditions. Let's call these expressions z1 and z2, and let z1 = (gx + h) and z2 = (jx + k). We need gj = a, (jh) + (gk) = b, and hk = c.

Applying this to our current problem, we have a=1, b=1, and c=-6. We need gj = 1. so g=1 and j=1 seem like a good guess for these values for now. This means that we need h+k = 1 and hk = -6.

Two numbers that fulfil these conditions are h=3 and k=-2.

This means that we can write the formula for the curve as (x)(x+3)(x-2). We know that the curve must intersect the x-axis when any on of these factors of the formula equal 0. This means that it must intersect the x-axis at x=0, x=3, and x=-3. 

Putting all this together, we can plot the curve as the 'curvy capital N' shape which intersects the x-axis at these points.