# Polynomials and Equations

When an expression is equated to another expression, we have an **equation**. For example,

\[\begin{array}{l}2x + 3 = 7\\{x^2} + x + 1 = 3x + 2\\ax + by + cz + d = e\end{array}\]

are all examples of equations. By definition, an equation involves an equality.

*Solving an equation* means finding those values of the variables for which the two sides of the equation are equal, or in other words, the equation is *satisfied*. For example, the equation \(2x + 1 = 3\) has the *solution* \(x = 1\). The equation \({x^2} - 1 = 3\) has the solution(s) \(x = \pm \,2\). The equation \(x + 2y = 3z\)has infinitely many solutions, some of which are

\[\begin{array}{l}x = 1,\,\,y = 1,\,\,z = 1\\x = - 1,\,\,y = 5,\,\,z = 3\\x = 3,\,\,y = 6,\,\,z = 5\end{array}\]

The equation \({x^2} + 1 = 0\) has no solution in the set of Reals, since there is no real number *x* such that \({x^2} = - 1\). However, this equation *will* have a solution in the set of **Complex Numbers**.

Thus, we note that a given equation may not have any solution in a particular number system, but may have solutions in other number systems.