Answer:
[tex]f(x)=3x^3-6x^2-15x+30[/tex]
Or
[tex]f(x)=3(x-2)(x+\sqrt{5})(x-\sqrt{5})[/tex]
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 [tex]x=\sqrt{5}[/tex] and 2 with a leading coefficient of 3.
By the irrational root theorem of polynomials, [tex]x=-\sqrt{5}[/tex] is also a root of the required polynomial.
By the factor theorem, we can write the polynomial in factored form as:
[tex]f(x)=3(x-2)(x+\sqrt{5})(x-\sqrt{5})[/tex]
We expand, applying difference of two squares to obtain
[tex]f(x)=3(x-2)(x^2-5)[/tex]
We expand further using the distributive property to get:
[tex]f(x)=3x^3-6x^2-15x+30[/tex]
