As a plant grows, new root hair cells are formed from unspecialised cells. How does an unspecialised cell become a new root hair cell?

Root hairs appear in the root maturation zone and differentiate from unspecialised cells called trichoblasts.

Explanation:

Root hair arises as a small protrusion surrounded by a cell wall, they are found in the piliferous area; they can originate in unspecialised epidermal cells, in some called trichoblasts. The bump begins to elongate and moves away from the epidermis, forming root hair. The root hairs as they lengthen, it is seen that their content shows a polarized distribution, the apical part is constituted mainly by vesicles secreted by the Golgi apparatus that contain the precursors of the cell wall that are to be released by exocytosis. Trichoblasts can be arranged in different ways, there are roots where any of the cells that are part of the rhizodermis can give rise to a root hair. In other roots, trichoblasts and atricoblasts have been seen to be distinguished alternately along a row of cells.

the answer is cell division : )