1. The problem is to find the points of intersection between the curves given by the equations $y = x^3$ and $y = 4x$. 2. To find the intersection points, set the two expressions for $y$ equal to each other: $$x^3 = 4x$$ 3. Rearrange the equation: $$x^3 - 4x = 0$$ 4. Factor out $x$: $$x(x^2 - 4) = 0$$ 5. Further factor the quadratic term: $$x(x - 2)(x + 2) = 0$$ 6. Set each factor equal to zero to find the roots: - $x = 0$ - $x - 2 = 0 \Rightarrow x = 2$ - $x + 2 = 0 \Rightarrow x = -2$ 7. Substitute each $x$ value back into one of the original equations (e.g., $y = 4x$) to find the corresponding $y$ values: - For $x = 0$: $y = 4(0) = 0$ - For $x = 2$: $y = 4(2) = 8$ - For $x = -2$: $y = 4(-2) = -8$ 8. Therefore, the points of intersection are: $$ (0, 0), (2, 8), (-2, -8) $$ These are the points where the cubic curve and the line meet.