A polynomial is a mathematical expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and exponentiation to non-negative integer powers. The exponents of the variables must be whole numbers (non-negative integers), and no division by a variable is allowed. From examples commonly given, the expression that qualifies as a polynomial is the one where all variable exponents are non-negative integers and no variable appears in a denominator or under a root. For instance, from common multiple-choice polynomial identification questions:
- An expression like x22−2x2\frac{x^2}{2}-\frac{2}{x^2}2x2−x22 is not a polynomial because it has a negative exponent when rewritten as x−2x^{-2}x−2.
- An expression like 2x−1\sqrt{2x}-12x−1 is not a polynomial because the exponent here is a fraction (1/2).
- An expression like x2+3x3/2/xx^2+3x^{3/2}/\sqrt{x}x2+3x3/2/x can be simplified to have whole number exponents in the terms. For example, simplifying 3x3/2/x=3x3/2−1/2=3x13x^{3/2}/\sqrt{x}=3x^{3/2-1/2}=3x^{1}3x3/2/x=3x3/2−1/2=3x1, which is a whole number exponent. Thus, it can be a polynomial depending on the simplification.
- Expressions like x−1x+1\frac{x-1}{x+1}x+1x−1 are not polynomials as they represent rational functions involving division by variables.
Therefore, the polynomial among typical options is the one where after simplification, the expressions only involve variables with non-negative integer exponents and do not involve division by variables or fractional exponents. In summary, a polynomial is an expression with terms made of variables raised to whole number exponents, multiplied by coefficients, combined using addition, subtraction, and multiplication only.
