Intermediate Algebra/Polynomials

A monomial of one variable, x, is an algabraic expression of the form

$c{x}^m$    

where

• $c$ is a constant, and
  
• $m$ is a non-negative integer (e.g., 0, 1, 2, 3, ...).

The integer $m$ is called the degree of the monomial.

A polynomial of one variable, x, is an algebraic expression that is a sum of one or more monomials. The degree of the polynomial is the highest degree of the monomials in the sum. An polynomial $P(x)$ can generically be expressed in the form

$P(x)=a_n{x}^n+...+a_i{x}^i+...a_2{x}^2+a_{1}x+a_0$

The constants a_i are called the coefficients of the polynomial.

Each of the individual monomials in the above sum, whose coefficient a_i ≠ 0, is called a term of the polynomial. When i = 0, xⁱ = 1 and the corresponding term simply equals the constant a₀. Also when i = 1, the corresponding term equals a₁ x.

A polynomial having two terms is called a binomial. A polynomial having three terms is called a trinomial.

Polynomial Equations

If a polynomial P(x) of at least degree 1 is set equal to 0 as follows:

$P(x)=0$

this results in an equation which may have one or more roots (solutions), i. e. number(s) whose value(s) for x make the equation true. These roots are called the zeroes of the polynomial (singular is zero). A polynomial of degree 1, a linear function, will always have 1 real zero. A polynomial of degree 2, a quadratic function, can have 0, 1, or 2 real zeroes. A polynomial of degree 3 (a cubic function) can have 1, 2, or 3 real zeroes. A polynomial of degree 4 can have 0, 1, 2, 3, or 4 real zeroes. In general, a polynomial of degree n, where n is odd, can have from 1 to n real zeroes. A polynomial of degree n, where n is even, can have from 0 to n real zeroes.

Solving Polynomial Equations

Some polynomial equations can be solved by factoring, and all equations of degrees 1-4 can be solved completely by formulae. Above degree 4, there are no formulae for solving completely, and you must rely on numerical analysis or factoring.

Rational roots of polynomial equations

If we have a polynomial P(x)

$P(x)=a_n{x}^n+...+a_i{x}^i+...a_2{x}^2+a_{1}x+a_0$

The only possible rational roots (roots of the form p/q) are in the form

$\frac{p}{q}=\frac{\text{factor}\text{of}{a}_n}{\text{factor}\text{of}{a}_0}$

External links

• Online interactive exercises on polynomials.

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