Cubic and Quartic Equations

1. Cubic and Quartic Equations
1. When the polynomial is a cubic expression in , the equation is called a cubic equation in . Similarly, when is a quartic expression in , the equation is called a quartic equation in .
2. Solving Cubic and Quartic Equations
   The equation is solved by factoring and using the following properties:
1. If , then or .
2. If , then , , or .
3. If , then , , , or .


2. Solving Cubic and Quartic Equations Using the Factor Theorem
   In the equation , if , then has as a factor. That is,
   where can be found using synthetic division.
   
   
3. Various Methods for Solving Quartic Equations
1. Quartic Equations with Common Terms
   If the equation has common terms, replace the common part with a single variable and factor it to solve.
2. Biquadratic Equations (of the form )
1. Replace with and factor the equation to solve.
2. Separate the quadratic term and transform it into the form to factor and solve.
3. Reciprocal Equations (of the form )
   An equation where the coefficients of each term are symmetric around the middle term is called a reciprocal equation. Quartic reciprocal equations can be solved as follows:
1. Divide both sides by , then replace with .
2. Find the value of , substitute it back into , and solve for .