1. **State the problem:** We need to find the cubic function $f(x)$ such that: - $f(x)$ touches the x-axis at the origin $(0,0)$. - $f(x)$ cuts the x-axis at $x=4$. - $f(x)$ passes through the point $(10,120)$. 2. **Understand the conditions:** - Touching the x-axis at $x=0$ means $x=0$ is a root of multiplicity 2 (the curve just touches and does not cross). - Cutting the x-axis at $x=4$ means $x=4$ is a root of multiplicity 1. 3. **Form the general cubic function:** Since $0$ is a root of multiplicity 2 and $4$ is a root of multiplicity 1, the function can be written as: $$f(x) = a x^2 (x - 4)$$ where $a$ is a constant to be determined. 4. **Use the point $(10,120)$ to find $a$:** Substitute $x=10$ and $f(10)=120$: $$120 = a \times 10^2 \times (10 - 4) = a \times 100 \times 6 = 600a$$ 5. **Solve for $a$:** $$a = \frac{120}{600} = \frac{1}{5}$$ 6. **Write the final function:** $$f(x) = \frac{1}{5} x^2 (x - 4) = \frac{1}{5} (x^3 - 4x^2) = \frac{1}{5} x^3 - \frac{4}{5} x^2$$ **Final answer:** $$f(x) = \frac{1}{5} x^3 - \frac{4}{5} x^2$$