Under what conditions is the product of two binomials a binomial?

Question

Under what conditions is the product of two binomials a binomial?

Answer 1

The product of two binomials will be a binomial if and only if the two are conjugates.

(a + b)(c + d) = ac + ad + bc + bd

If ac + ad = 0

c = - d

and the 2nd term is no longer a binomial.

If ac + bc = 0

a = - b

and the 1st term is no longer a binomial

If ac + bd = 0

ac = - bd

(a + b)(c + d) = ad + bc

c = - bd/a

(a + b)(- bd/a + d) = ad - b^2d/a

(a + b)(- bd + ad)/a = (a^2d - b^2d)/a

(a + b)(- b + a)d/a  = (d/a)(a^2 - b^2)

(a + b)(a - b) = (a^2 - b^2)

In the same way, one can show that the only other product which is a binomial is

(c + d)(c - d) = c^2 - d^2