Question

write a polynomial function with rational coefficients so that p(x)=0 has the given roots 2,3,5

Answer 1

A polynomial has a root \(q\) if (and only if) \((x-q)\) is a factor of that polynomial. So a polynomial with roots 2, 3 and 5 must have the factors \((x-2),(x-3)\) and \((x-5)\). So the simplest such polynomial you just get by multiplying these:
\[p(x)=(x-2)(x-3)(x-5)\]
Now it only remains to multiply everything out to get it on the form that the question asked.