What is the significance of Dasharatha's marriage with three women and why was lord Rama born into a family where three mothers were there?
What is the significance of Dasharatha's marriage with three women
King Dasaratha of Kosala has three wives, all of whom are unable to conceive (Sivaraman 107). His first and oldest wife, Kausalya, has rights to the throne for her son unlike his second and third wives, Sumitra, and Kaikeyi. The love King Dasaratha has for Kaikeyi is comparable to the love his father, Aja had for his mother, Indumati (Madan 191). King Dasaratha’s first two wives are unable to have children, so they are unable to provide a successor to the throne. King Dasaratha believes Kaikeyi is able to conceive and thus promises her father, Aswapati, that her son would be the kingdom’s next king. However, eventually all of King Dasaratha’s wives have sons and due to the seniority of Kausalya as first wife, her son is announced as King Dasaratha’s successor. When Kaikeyi learnt of King Dasaratha’s promise to her father, she asked King Dasaratha to grant her two boons (promises) she had earned when she saved his life.
So clearly being a King he wanted an heir to the throne and when 1st wife wasn't able to bear any child he married 2nd one and when 2nd one wasn't able to bear any child, he married 3rd one.
why was lord Rama born into a family where three mothers were there?
Above states why Dashratha had 3 wives.
As King Dasaratha was originally unable to have children, he reached out to the gods by performing an Asvamedha, the horse sacrifice, asking them to bestow a child upon him. Collectively, many gods pressured Lord Visnu into manifesting himself into the sons of King Dasaratha in order to defeat the demon, Ravana. However, there is a disagreement over how King Dasaratha received the potion that would ultimately lead to the birth of his sons. Some say Visnu himself presented the potion to King Dasaratha during the sacrifice (Sivaraman 107), while others argue that Agni, the god of fire, presented King Dasaratha with Caru, a sacrificial food (Madan 191). Despite the disagreement on how King Dasaratha gained this magical substance, all three of his wives received portions of it. Instructed to divide the potion between his wives; King Dasaratha gave half to Kausalya due to her seniority and the other half to Kaikeyi due to his fondness for her. Unfortunately, this did not leave any for Sumitra which caused Kausalya and Kaikeyi to each give her half of their portions. Since Sumitra technically received two servings, she bore two sons (Madan 191). Kausalya bore Rama, Sumitra bore twins, Laksmana, and Satrughna, and Kaikeyi bore Bharata.
So above explains why and how it happened. Refer
It was common in all Hindu kings to have multiple wives.
Dasaratha had multiple wives but he had no heirs for a long time. Messenger of God (Rushi) met Dasaratha and he performed a Yoga to have heirs. The Yoga Prasad(food) half (food) was given to his first wife Kausalya, one sixth to his youngest wife Kaikeyi and the rest to Sumitra, his middle wife. Eventually they all gave birth: Kausalya to Rama, who was one half of Vishnu, Kaikeyi to Bharata who was one sixth of Vishnu and Sumitra to twins, Lakshmana and Satrughna, who were both one sixth of Vishnu. So Lord Rama was born to Dasaratha and had 3 mothers.
P.S : as per Hindu Marriage Act 1955 its illegal to have multiple wives as a Hindu, link : https://en.wikipedia.org/wiki/The_Hindu_Marriage_Act,_1955
you can find people who have mutliple wives who were married before 1955