Question

Give an example of a polynomial which is a binomial of degree \[20\].

Answer 1

Hint: A polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. A polynomial containing two terms is called a binomial. Degree of a polynomial is the highest power of a variable in that polynomial.

Complete answer:
In the question we are asked to write a polynomial that should be a binomial of degree \[20\]. It means the polynomial should contain two terms and the highest power of variable should be \[20\]. There can be many examples with these given conditions.
So, the required polynomials are;
\[{x^{20}} + 5x,{x^{20}} + 1,{x^{20}} - 5{x^3}\]
Here we can see that there are only two terms in each of the examples and the highest power in each is \[20\].

Additional details:
The \[x\] occurring in a polynomial is commonly called a variable or an indeterminate. When the polynomial is considered as an expression, \[x\] is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the function defined by the polynomial, then \[x\] represents the argument of the function, and is therefore called a "variable". A polynomial \[P\] in the indeterminate \[x\] is commonly denoted either as \[P\] or as \[P\left( x \right)\].

Note: One major mistake that many students make is that while classifying a given expression as a polynomial, they do not see whether the exponents of the terms in the expression are non-negative or not. If the exponent of any of the terms in the given expression is negative then that expression cannot be considered as a polynomial.