Write down the denominator of the transfer function and equate it to zero. Use the transfer function 10x/(x^2 + 13x + 42) for this example. Since the denominator is x^2 + 13x + 42 equating the denominator to zero yields x^2 + 13x + 42 = 0.
Factor or algebraically simplify the denominator equation so that the unknown variable in the denominator can be solved. In this example, the equation in the denominator can be factored as (x+6)(x+7) since (x+6)(x+7) is equivalent to x^2 + 13x + 42
Solve the factored or algebraically simplified equation for the values that will make it equal to zero; (x+6)(x+7) = 0. For this example setting x+6 and x+7 = 0 to zero and solving will let you determine the values of x that will make the denominator zero. Solving x + 6 = 0 results in a value of x of -6. Solving x + 7 = 0 results in a value of x of -7.
Check that the poles you calculated will result in a value of 0 in the denominator. Substitute each pole value you obtained into the original denominator equation. Substituting -6 into x^2 + 13x + 42 produces 36 + -78 + 42 which equals 0. Substitute -7 into x^2 + 13x + 42 to obtain 49 - 91 + 42 which equals 0.
Conclude that the poles of the transfer functions are -6 and -7 since these are the values that make the denominator of the transfer function equal to 0.