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Polynomials Introduction,
Examples of Polynomial Expressions


A polynomial expression in one variable is of the following standard form.

    a_{n}x^n \space + \space a_{n{\text{–}}1}x^{n{\text{–}}1} \space + \space …… \space + \space a_{2}x^2 \space + \space a_{1}x \space + \space a_{0}           ( where   a_{n} \space {\cancel{=}} \space 0 )


The terms in a polynomial can be constants, or variables with a positive whole number exponent.




Parts of a Polynomial


We can observe a standard polynomial expression to identify the different parts that make up the polynomial.

Variable:

x  is a ‘variable’, which represents a numerical value that can change.

A range of letters or symbols can be used for variables, but most commonly seen is the letter  x.
There can also be more than one variable present in a polynomial.

A variable in a polynomial must have an exponent that is non negative and a whole number.

Terms such as   x^{\tt{{\text{-}}3}}\sqrt{x}   and   x^{1/4}   are not allowed in polynomials.


Coefficient:

5  and  2  are coefficients.
Numbers that exist alongside and multiply a variable.

Unlike variables, coefficients are allowed to have fractional exponents, and can be fractions also.

Coefficients like   \sqrt{5}   and  \frac{1}{2}   are perfectly fine.


Constant:

3  is a constant, and unlike a variable is always the same value in the polynomial.





Degree of a Polynomial

The degree of a polynomial in one variable, is the largest exponent of that variable that is present.


2x^{\tt{3}} + 4x \space {\text{–}} \space 5    is a polynomial expression of degree “3“.

x^{\tt{2}} \space {\text{–}} \space 3x + 4    is a polynomial expression of degree “2“.


The degree of a polynomial is an indication of how many roots/solutions the polynomial has.

A polynomial of degree  “2”  has at most 2 solutions.
A polynomial of degree  “3”  has at most 3 solutions,  and so on.


Polynomial Names:

Polynomials can have specific names depending on the degree.

  –  A linear polynomial is of degree 1, for example     x + 6.
  –  A quadratic polynomial is of degree 2, for example     x^{\tt{2}} + 2x + 3.
  –  A cubic polynomial is of degree 3, for example     x^{\tt{3}} \space {\text{–}} \space x + 7.


There are names for larger degree polynomials also but the three listed above are usually the most common to encounter.


A constant is also a type of polynomial, if you have just a number such as  4,
this is technically a polynomial of degree  “1”.

There doesn’t always need to be a variable present.


It should also be mentioned that degree can sometimes be referred to as ‘order’.






Examples of Polynomial Expressions,
Equations and Functions



Polynomial Equations:

When a polynomial is presented in the form   x^{\tt{2}} + 4x + 2,
it is a polynomial expression.


But we can also write as a polynomial equation or function.
A polynomial equation is when a polynomial is set equal to a certain value, even zero.


x^{\tt{2}} + 4x + 2 = 0    and    x^{\tt{2}} + 4x + 2 = 9   are both examples of a polynomial equation.


We can solve many polynomial equations and find the roots/solutions.

We see how this can be done in further sections, but as mentioned earlier in this page, a polynomial equation will not have more solutions than what the degree of the equation is.

So    x^{\tt{2}} + 4x + 2 = 0    and    x^{\tt{2}} + 4x + 2 = 9    would both have at most 2 solutions.




Polynomial Functions:


A polynomial can presented and written as a polynomial function.

Where we could solve for a range of values that could be input into the polynomial.


f(x) = x^{\tt{2}} + 4x + 2     and     f(x) = 4x^{\tt{3}} \space {\text{–}} \space x^2 + 4x + 2

are examples of polynomial functions.





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