The transmission of genetic traits in polyploids is more difficult to calculate than in diploids because a gene for a recessive trait in a triploid, for example, would have to appear in the same location on all three of its homologous chromosomes in order for it to be phenotypically apparent. Such calculations, when done for diploids, rely upon binomial equations and generate a familiar ratio of 9AB:3aB:3Ab:1ab, whereas the calculations for polyploid plants require the use of trinomial equations for triploids and quadrinomial equations for tetraploids, instead of the traditional binomial (A + B)2 that generates the familiar 9AB:3aB:3Ab:1ab ratio for diploids. Thus for a trinomial (three gene) the equation will be the expansion of (A + B + C)3.
The use of polyploids in laboratory research has allowed research into the function of specific genes. For instance, triploid female fruit flies crossed to diploid males were used to create a diploid offspring with a chromosome of a sibling species. In this experiment, the tiny fourth chromosome of Drosophila simulans was inserted into an otherwise diploid D. melanogaster offspring. This permitted analysis of the genes shared in common (most of them) as well as gene differences that led to visible malformations in the hybrid fly. Triploid flies have also been crossed to irradiated diploid males to prove that X rays induce breaks in chromosomes, causing apoptosis and embryonic abortion.
Muller, H. J. "Why Polyploidy is Rarer in Animals than in Plants." The American Naturalist LIX (1925): 346-353.
Dobzhansky, Theodosius. "Patterns of Evolution." In Genetics and the Origin of Species,Theodosius Dobzhansky, ed. New York: Columbia University Press, 1951.