 Hello and welcome to the session. In this session, we will discuss how to extend polynomial identities to the complex numbers. Now let us recall that a complex number is of the form a plus iota b where a is the real part and b is the imaginary part of the complex number and iota is equal to square root of minus 1. Now we know that the real identity x square minus y square is equal to x minus y the whole into x minus y the whole. Now when we see how can we write polynomial of type plus y square will improve that x square plus y square is equal to x plus iota y the whole into x minus iota y the whole. Now let us take the left hand side of this equation which is x square plus y square. Now this can be written as x square minus of minus 1 the whole into y square because we know that minus of minus 1 is plus 1. Now we know that iota is equal to square root of minus 1 therefore iota square is equal to minus 1. So when we will write minus 1 as iota square and this will be equal to x square minus iota square y square further this can be written as x square minus iota y whole square. Now this is from the form a square minus b square here a is x and b is iota y so this can be written as a plus b the whole into a minus b the whole So this can be written as x plus iota y the whole into x minus iota y the whole right hand side of this equation. So when we write plus y square is equal to x plus iota y the whole into x minus iota y the whole then simplify right hand side of this equation that is x plus iota y the whole into x minus iota y the whole now this is equal to x into x minus iota y the whole plus iota y into x minus iota y the whole now let us open the brackets so this is equal to x square minus iota xy plus iota xy minus iota square y square now solving this is equal to minus iota square y square now we do that iota square is equal to minus 1 so this is equal to x square minus of minus 1 the whole into y square which is equal to x square now minus of minus 1 is plus 1 so we can write it as x square plus y square which is equal to the left hand side of the given equation so we have proved that x square minus y square is equal to x plus iota y the whole into x minus iota y the whole now consider the following quadratic equation that is x square plus 4 is equal to 0 now we want to find its solution now this equation can be written as x square minus of minus 4 is equal to 0 this implies x square minus of 4 into minus 1 the whole is equal to 0 now we know that iota square is equal to minus 1 so further we can write this as x square minus of now 4 can be written as 2 square into minus 1 can be written as iota square the whole is equal to 0 this implies x square minus 2 iota whole square is equal to 0 now using this formula this implies x square minus 2 iota the whole into x minus 2 iota the whole is equal to 0 so either x plus 2 iota is equal to 0 or x minus 2 iota is equal to 0 which implies x is equal to minus 2 iota or x is equal to 2 iota thus this quadratic equation has two imaginary roots ultimately we can directly use this identity to solve the above quadratic equation now let us solve the above quadratic equation using this identity where we write x square plus 4 is equal to 0 which implies x square plus now 4 can be written as 2 square is equal to 0 now this is of the form x square plus y square whereby is 2 so using this identity we have x square plus 2 square is equal to x plus 2 iota the whole into x minus 2 iota the whole is equal to 0 which implies x is equal to minus 2 iota or x is equal to 2 iota so directly using this identity we have obtained the two roots that is the two imaginary roots of this quadratic equation now note that the complex number and its conjugate always occur in pair it means if a complex number is a root of any polynomial then its conjugate will also be its root so we can factorize a polynomial into linear factors using above identity for example factorize 3x cube plus 9x now let us take 3x common from both these terms so this is equal to 3x into x square plus 3 iota which is equal to 3x into x square plus now 3 can be written as square into 3 whole square the whole now again using this identity here we have x square plus root 3 whole square is equal to x plus root 3 iota the whole into x minus root 3 iota the whole so this is equal to 3x into x plus root 3 iota the whole into x minus root 3 iota the whole so this is the required factorization of the given polynomial session we have discussed how can we use complex numbers in solving polynomials and this completes our session hope you all have enjoyed the session