Chapter 2: Polynomials

Ans:  Here we have given an expression  $ 8 $ in which the degree of variable is zero. Therefore, it is a constant Polynomial.

Ans: $ \sqrt 3 {x^2} - 2x $ 

Here, in each term the power of x is a whole number. Therefore, this expression is a Polynomial.

Ans: $ 1 - \sqrt {5x}  $ 

We can also write it as  $ 1 - \sqrt 5 {x^{\dfrac{1}{2}}} $ . Here the degree of variable is not a whole number. Therefore, this expression is not a Polynomial.

Ans: $ \dfrac{1}{{5{x^{ - 2}}}} + 5x + 7 $ 

We can also write it as  $ \dfrac{{1{x^2}}}{5} + 5x + 7 $ . Here in this expression each variable term has the power of x in a whole number. Therefore, this expression is a Polynomial.

Ans: $ \dfrac{{\left( {x - 2} \right)\left( {x - 4} \right)}}{x} $ 

We can also write it as 

 $ \dfrac{{{x^2} - 6x + 8}}{x} = x - 6 + \dfrac{8}{x} = x - 6 + 8{x^{ - 1}} $ 

Here, the exponent of variable x in the third term  $ 8{x^{ - 1}} $  is – 1, and this is not a whole number.

 Therefore, this algebraic expression is not a Polynomial.

Ans: $ \dfrac{1}{{x + 1}} $ 

We can also reduce it to  $ {\left( {x + 1} \right)^{ - 1}} $ , which cannot be reduced to an expression in which the exponent of the variable x had only whole numbers in each of its terms. Therefore, this algebraic expression is not a Polynomial.

Ans: $ \dfrac{1}{7}{a^3} - \dfrac{2}{{\sqrt 3 }}{a^2} + 4a - 7 $ 

In this expression, the exponent of a in each term is a whole number, so this expression is a Polynomial.

Ans: $ \dfrac{1}{{2x}} $ 

We can also write it as  $ \dfrac{1}{2}{x^{ - 1}} $ . Here, the exponent of the variable x is   $  - 1 $  , which is not a whole number so, this algebraic expression is not a Polynomial.

Ans: The given statement is False because binomials have exactly two terms.

Ans: Every Polynomial cannot be a binomial because a Polynomial had many terms but a binomial had only two terms in the expression. For example,  $ {x^4} + {x^3} + {x^2} + x + 1 $ is a Polynomial but not a binomial. Therefore, the given statement is false.

Ans: The given statement is True because a binomial is a Polynomial whose degree is a whole number and it is greater than or equals to one and contain two terms. For example,  $ {x^5} - 1 $  is a binomial of degree  $ 5 $ .

Ans: The given statement is False, because zero of Polynomial can be any real number.

Ans: The given statement is False, because a Polynomial can have any number of zeroes which depends on the degree of the Polynomial.

Ans: The given statement is False. For example, consider the two Polynomials  $  - {x^5} + 3{x^2} + 4 $  and  $ {x^5} + {x^4} + 2{x^3} + 3 $ .The degree of each of these polynomials will be 5. Their sum is  $ {x^4} + 2{x^3} + 3{x^2} + 7 $ . The degree of this Polynomial is  $ 4 $  not  $ 5 $ .