Let \(d\mu_1\) and \(d\mu_2\) be two measures with the same support \(E\). They are said to be related to each other by a first degree polynomial multiplication, if \[ (x-q)d\mu_1(x)=cd\mu_2(x), \] where the first degree polynomial \(c^{-1}(x-q)\) is positive on \(E\). The authors study the connection between two sequences of orthogonal polynomials and sequences of recurrence coefficients. The results are applied to certain problems regarding Sobolev orthogonal polynomials associated with pairs of measures.