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} = \not 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.
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.
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.
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.
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.