Programmers and Teachers:
Lesson on Solving Polynomials
How to Solve for a Polynomial Variable
When solving for a variable within a polynomial equation, we work algebraically to isolate it. To isolate a variable, we use the reverse order of operations to move all terms and numbers to the opposite side of the equation of the variable. Once the target variable is alone on one side of the equation, it is solved.
Solving Variables in Special Polynomials
Some polynomial equation variables cannot be solved via basic isolation techniques. For these special polynomials, we may use a variety of other solving techniques.
Commonly used techniques are factoring and the quadratic formula. Factoring may be used when the variable has an exponent. The quadratic formula may be used for second-degree polynomials.
Sometimes a polynomial does not have any real, whole number, fractional, or rational solutions. When this happens, we may employ a computer that solves using numerical computation. The calculator on this page uses numerical computation for these special case polynomials.
How to Factor Polynomials
Factoring a polynomial is effectively the reverse action of simplifying terms grouped by parenthesis. For any factorable polynomial, we may use a method called completing the square (see our lesson for full tutorial). A polynomial must be in an equation to complete the square.
If we are simply factoring a polynomial for the sake of reaching factored form, we are finished once the square is completed. However, completing the square also enables us to determine the zeroes or “roots” of an equation by converting it to a factored form if we desire a solution to a variable.
How the Calculator Works
The CAS is fed your polynomial and whether you are solving for x or factoring. The CAS treats the computation symbolically, preserving exact values of variables and numbers. In special cases where there are no rational or real number solutions, the CAS uses numerical methods to achieve a very accurate, approximated solution.
Once your answer is calculated, it is converted to LaTeX code. LaTeX is a math markup and rendering language that allows for graphical equation printing on webpages. This page’s local LaTeX script renders that code in the answer area as the solution you see.