Find all values of 𝑐 such that the domain of 𝑓(𝑥) = 𝑥+3/𝑥^2+3𝑐𝑥+6 is the set of all real numbers.

Answer 1

x + 3/x^2 + 3cx + 6 will always exclude x=0 from the domain, as it is impossible to divide by zero

no value of c can change that, as long as x^2 is in the denominator. 0 is a real number, so the domain will never be the set of all real numbers regardless of the value of c, if c is a constant. Using c is a common way to indicate a constant, as the word constant begins with "c".

But if c = -1/x^3 then the 4 term polynomial becomes

x + 3/x^2 +3(-1/x^3)x + 6

x + 3/x^2 - 3/x^2 + 6

= x+6, whose domain is any real number