Introduction to Polynomial Equations
The Polynomial Equations are numerical equations that can be reduced to the form P(x) = 0, where P(x) is a polynomial.
A polynomial equation has only numerical coefficients, and the unknown never appears in the denominator of any fraction.
In this lesson we will establish some fundamental concepts about polynomial equations; in the rest of the chapter we will focus on linear (first-degree) equations.
- A polynomial equation is an equation that can be reduced to the form P(x) = 0, where P(x) is a polynomial;
- A polynomial equation is in normal form if the polynomial is in normal form;
- The degree of a polynomial equation equals the degree of its polynomial.
Polynomial equations
In the previous lesson we studied the concept of equivalent equations. In short, two equations are equivalent if they have the same solutions. We also studied the principles of equivalence that allow us to transform a given equation into another equivalent equation solved by the same solutions.
Now we can use these two concepts to introduce the notion of a polynomial equation.
Given a numerical and integer equation — i.e. one in which the unknown does not appear in any denominator and all coefficients are numerical — it can be shown that it is equivalent to an equation of the form
in which the left-hand side is a polynomial.
To understand how this is possible, let us study the following example:
Consider the equation
By moving all constant and variable terms to the left (using the first principle of equivalence) we obtain
Carrying out the calculations gives
On the left we have indeed obtained a polynomial.
Such equations are called Polynomial Equations:
Polynomial equation
A Polynomial Equation is a numerical algebraic equation that can be reduced to the form
where
The term
Polynomial equations are of fundamental importance in mathematics because they frequently arise in solving problems across many disciplines.
Normal form of an equation
Because a polynomial equation can be reduced to an equality between zero and a polynomial, it is always possible to write the polynomial in normal form.
Take the equation
Since it is a numerical equation, it is also a polynomial equation. We can put the polynomial into normal form; first expand the left-hand side:
Now move the term 9 to the left (using the “transport rule”) and combine like terms:
The polynomial on the left is now in normal form.
Since a polynomial can always be put in normal form, we can define a normal form for polynomial equations:
Normal form of a polynomial equation
The Normal Form (or Canonical Form) of a polynomial equation is an equivalent equation of the form
where the polynomial
Degree of an equation
To a polynomial equation in canonical form we can associate the degree of its polynomial.
Degree of an equation
The Degree of an Equation is the degree of the polynomial in normal form that defines it, provided it can be reduced to a polynomial equation.
The degree is a very important piece of information about an equation. Indeed, the Fundamental Theorem of Algebra states that an equation of degree
Linear equations
The simplest polynomial equations are first-degree equations, also called linear equations:
Linear equation
A Linear Equation is a first-degree polynomial equation:
They are called “linear” because in analytic geometry they are used to represent a straight line on the Cartesian plane (hence a “line”). We will study the equation of a straight line in geometry lessons.
In this chapter of the mathematics guide we will focus on solving first-degree or linear equations.
In summary
The aim of this lesson has been to set out some concepts regarding numerical equations, starting from the principles of equivalence.
We have seen what a polynomial equation is, what the normal form of an equation is, and, above all, what the degree of an equation is.
In the next lesson we will focus on first-degree equations and see how to solve them.