Answer:
![f(x)=3x^3-6x^2-15x+30](https://tex.z-dn.net/?f=f%28x%29%3D3x%5E3-6x%5E2-15x%2B30)
Or
![f(x)=3(x-2)(x+\sqrt{5})(x-\sqrt{5})](https://tex.z-dn.net/?f=f%28x%29%3D3%28x-2%29%28x%2B%5Csqrt%7B5%7D%29%28x-%5Csqrt%7B5%7D%29)
Step-by-step explanation:
For a polynomial function of lowest degree with rational real coefficients, each root has multiplicity of 1.
The polynomial has roots
and 2 with a leading coefficient of 3.
By the irrational root theorem of polynomials,
is also a root of the required polynomial.
By the factor theorem, we can write the polynomial in factored form as:
![f(x)=3(x-2)(x+\sqrt{5})(x-\sqrt{5})](https://tex.z-dn.net/?f=f%28x%29%3D3%28x-2%29%28x%2B%5Csqrt%7B5%7D%29%28x-%5Csqrt%7B5%7D%29)
We expand, applying difference of two squares to obtain
![f(x)=3(x-2)(x^2-5)](https://tex.z-dn.net/?f=f%28x%29%3D3%28x-2%29%28x%5E2-5%29)
We expand further using the distributive property to get:
![f(x)=3x^3-6x^2-15x+30](https://tex.z-dn.net/?f=f%28x%29%3D3x%5E3-6x%5E2-15x%2B30)