Polynomials and Equations

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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.

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