Solving Systems of Polynomials

Study Notes

Solving Systems of Polynomials

Definition

A system of polynomial equations is a set of two or more polynomial equations in two or more variables.
The goal is to find the values of the variables that satisfy all the equations in the system.

Methods for Solving Systems

Substitution Method
- Solve one equation for one variable.
- Substitute the expression into the other equation.
- Solve the resulting equation for the other variable.
Elimination Method
- Multiply the equations by necessary multiples to make the coefficients of one variable opposites.
- Add the equations to eliminate one variable.
- Solve the resulting equation for the other variable.
Graphing Method
- Graph the equations on the same coordinate plane.
- Find the point of intersection, which represents the solution to the system.

Key Concepts

Independent and Dependent Systems
- Independent systems: have a unique solution.
- Dependent systems: have infinitely many solutions.
Inconsistent Systems
- No solution exists.
- The lines or curves do not intersect.
Equivalent Systems
- Two systems with the same solution set.

Solving Systems with Polynomials

Substitution with Polynomials
- Solve one equation for one variable, typically the simpler equation.
- Substitute the expression into the other equation, which may contain polynomials.
- Factor the resulting equation, if possible, to find the solutions.
Elimination with Polynomials
- Multiply the equations by necessary multiples to make the coefficients of one variable opposites.
- Add the equations to eliminate one variable, which may result in a polynomial equation.
- Factor the resulting equation, if possible, to find the solutions.

Examples and Applications

Solving systems of linear and quadratic equations.
Modeling real-world problems, such as projectile motion, optimization, and physics.
Using systems to solve problems involving area, volume, and perimeter.

Description

Learn how to solve systems of polynomial equations using substitution, elimination, and graphing methods. Understand key concepts such as independent and dependent systems, inconsistent systems, and equivalent systems.