Question

Which of the following is a polynomial?
 $ \left( a \right)2x $
 $ \left( b \right){x^2} + {y^{ - 2}} - 2{z^2} $
 $ \left( c \right)5{x^3}{y^2}{z^3} $
 $ \left( d \right)x + {x^2} + {x^3} + {x^4} $

Answer 1

Hint: In this particular question use the concept that polynomials are an algebraic expression that contains one term or more than one term but the power of the variable is in the form of whole numbers so use these concepts to reach the solution of the question.

Complete step-by-step answer:
As we all know that a polynomial is an algebraic expression (i.e. x, 2x and so on) which contains one term or more than one terms for example:
\[\left( i \right)\] x, $ \left( {ii} \right)x + {x^2} $ and so on but the power of the x always in the form of a whole number.
A whole number is a number that does not contain negative numbers i.e. it only has positive numbers.
For example: 0, 1, 2.......
Whereas, -1, -2...... is not a whole number.
So a polynomial cannot be in the form of ( $ {x^{ - n}} $ ), where n is a positive integer starting from 1, it is always in the form of ( $ {x^n} $ ).
Now when we check the options only option (b) i.e. ( $ {x^2} + {y^{ - 2}} - 2{z^2} $ ) contains negative power of a variable so it never be a polynomial.
Rest of the options are in standard form of a polynomial so all of the options except option (b) are the correct answer.
So this is the required answer.
Hence options (a), (c) and (d) are the correct answer.

Note: Whenever we face such types of questions always remember that if a polynomial has only one term then the power of the variable cannot be zero otherwise it convert in to a single digit number which is 1, if the polynomial has more than one terms then the power of the variable can be zero except one term, so the general form of the polynomial is, $ a{x^3} + b{x^2} + cx + d $ , where a, b, c and d belongs to real numbers.