Any algebraic expression with two terms is called a binomial expression.
Example:
\(2x+3y\), here we have two terms, they are \(2x\) and \(3y\). Thus, this expression is called as binomial expression.
Multiplying binomial with a binomial
Let us recall the distributive proportion.
If \(a\) is a constant, \(x\) and \(y\) are variables, then \(a(x + y) = ax + ay\).
Example:
If the dimensions of the rectangle are \(2x + 3y\) and \(5x + y\), then its area is computed by the product of the given binomials.
This can be written as .
Applying the distributive property,
\(= (2x + 3y)5x + (2x + 3y)y\)
\(= 10x^{2} + 15xy + 2xy + 3y^{2}\)
\(= 10x^{2} + 17xy + 3y^{2}\)
Multiplying binomial with a trinomial
Example:
Find the product of and .
Applying the distributive property,
\(=\) \((3×2)\) \(p^{1+1}q^{3}\) \(+\) \(2\) \(pq^{1+3}\) \(-\) \((3×5)\)\(p^{1+2}\) \(-\) \(5\) \(p^{2}q\) \(+\) \((3×3)pq^{4}\) \(+\) \((3)q^{1+4}\)
\(=\) \(6\) \(p^{2}q^{3}\) \(+\) \(2\) \(pq^{4}\) \(-\) \(15\)\(p^{3}\) \(-\) \(5\) \(p^{2}q\) \(+\) \(9pq^{4}\) \(+\) \(3q^{5}\)
Combine the like terms.
\(=\) \(6\) \(p^{2}q^{3}\) \(+\) \(11\) \(pq^{4}\) \(-\) \(15\)\(p^{3}\) \(-\) \(5\) \(p^{2}q\) \(+\) \(3q^{5}\)