algebra, polynomial functions

Question

how can you argue that f(x) = 3x^4+2x^2+5 has no factor of the form x - c, where c is a real number? and how can we solve this by using algebra?

Answer 1

Having a factor of the form that you write means that the polynomial f(x) has at least a real root. But as you can see 3x^{4}+2x^{2}+5 is greater than or equal to 5 (the exponents are even powers). So f(x) has no real roots. Therefore, f(x) has no factor of the form x-c for any c real number.