We consider the real polynomials of degree d + 1 with a fixed point of multiplicity d ≥,2. Such polynomials are conjugate to fa, d(x) = axd(x −,1) + x, a ∈,R \ {0}. In this family, the point 0 is always a non-hyperbolic fixed point. We prove that for given d, d ′, , and a, where d and d ′,are positive even numbers and a belongs to a special subset of R−, , there is a ′,< 0 such that fa, d is topologically conjugate to fa′, , d′, . Then we extend the properties that we have studied in case d = 2 to this family for every even d > 2.