Question

The difference between an algebraic expression and a polynomial is: \[\]
A. The exponents of polynomial terms are whole numbers while that of algebraic expression are not.\[\]
B. The exponents of algebraic expression terms are whole numbers while that of polynomial are not.\[\]
C. The constant term is absent in algebraic expression while it is present in polynomial.\[\]
D. None of the above\[\]

Answer 1

Hint: We recall the definition of polynomial and expression which both consist of variables, constants and exponents on constant. We recall that a polynomial must have whole number exponents on its variables but an algebraic expression may also have fractional or negative integral exponents. \[\]

Complete step by step answer:
We know from algebra that a constant is a known number and variable is unknown quantity. The variables are generally named as $x,y,z$.\[\]
We know that an algebraic expression is an expression made from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation). For example some algebraic examples are $5,5x+2,y\sqrt{x},\dfrac{1+{{x}^{2}}}{1-{{x}^{2}}},3xy+6x$ etc.\[\]
We know that a polynomial is an algebraic expression consisting of constants, variables and coefficients that involves only the operations of addition, subtraction, multiplication, and whole number exponents on the variables, for example some polynomials are $2,2x+3,2{{x}^{2}}+\dfrac{3}{4}x+9$ etc. \[\]
We see that all polynomials are algebraic expressions but all algebraic expressions may not be a polynomial. An algebraic expression becomes a polynomial when all the exponents on the variable terms are whole numbers and not in fractional expressions. For example the expressions $\sqrt{x}={{x}^{\dfrac{1}{2}}},{{x}^{-3}}=\dfrac{1}{{{x}^{3}}}$ are algebraic expressions but they are not polynomials because the exponents $\dfrac{1}{2},-3$ on the variable $x$ are fractions and negative integers. \[\]

So, the correct answer is “Option A”.

Note: We note that the highest exponent on a polynomial is called degree. The other name of the whole number is non-negative integer and all real numbers are polynomials with power 0 on variables. A polynomial with degree 1 is called linear polynomial, with degree 2 is called a quadratic polynomial and degree 3 is called a cubic polynomial. A polynomial with degree $n$ is called ${{n}^{\text{th}}}$ degree polynomial.