Chapter 2: Polynomials

Ans: (C) is the correct option.

Polynomials are an algebraic expression that consist of variables and coefficients. Here the power of variables should be in whole number like 1,2,3 etc.

 (A)  $ \dfrac{{{x^2}}}{2} - \dfrac{2}{{{x^2}}} $ 

Here the coefficient  $  - 2 $  is multiplied with  $ {x^{ - 2}} $ . So here the power of x is not a whole number. Hence, it is not a Polynomial.

(B) $ \sqrt {2x}  - 1 $ 

Here the coefficient  $ \sqrt 2  $  is multiplied with  $ {x^{\dfrac{1}{2}}} $ . So here the power of x is not a whole number. Hence, it is not a Polynomial.

(C)  $ {x^2} + \dfrac{{3{x^{\dfrac{3}{2}}}}}{{\sqrt x }} $ 

Here the coefficient  $ 1 $  is multiplied with  $ {x^2} $  and the coefficient  $ 3 $  is multiplied with \[{x^{\dfrac{3}{2} - \dfrac{1}{2}}} = {x^{\dfrac{2}{2}}} = x\]  . So here the power of x is a whole number in the whole equation. Hence, it is a Polynomial.

(D)  $ \dfrac{{x - 1}}{{x + 1}} $ 

It is not a standard format of a Polynomial. Hence, it is not a Polynomial.

Ans: Option (B) is correct.

Here  $ \sqrt 2  $  is a constant Polynomial. We can also write it as  $ \sqrt 2 {x^0} $ . Therefore, the degree of this Polynomial is  $ 0 $ .

Ans: Option (A) is correct.

The highest power of x in the Polynomial is called the degree of Polynomial. We have the term with the highest power of x is  $ 4{x^4} $ , which is  $ 4 $ .

Therefore, the degree of Polynomial is  $ 4 $ .

Ans: Option (D) is the correct answer.

Like any constant value, the value 0 can be considered as a (constant) Polynomial, called the zero Polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either. Therefore, the degree of the zero Polynomial is not defined.

Ans: The correct option is (B).

We have,

 $ p\left( x \right) = {x^2} - 2\sqrt 2 x + 1 $ 

Here, we have given the value of  $ x = 2\sqrt 2  $ .

 So, on putting the value in above equation, we get

 $ p\left( {2\sqrt 2 } \right) = {\left( {2\sqrt 2 } \right)^2} - 2\sqrt 2 \left( {2\sqrt 2 } \right) + 1 $ 

 $ p\left( {2\sqrt 2 } \right) = 4 \times 2 - 4 \times 2 + 1 $ 

 $ p\left( {2\sqrt 2 } \right) = 8 - 8 + 1 $ 

 $ p\left( {2\sqrt 2 } \right) = 1 $ 

Ans: The correct option is (A).

Here we have given a Polynomial  $ 5x - 4{x^2} + 3 $ .

In this we have to put  $ x =  - 1 $ , and then we have to find the value of Polynomial

Therefore,

 $ p\left( x \right) = 5x - 4{x^2} + 3 $ 

 $ p\left( { - 1} \right) = 5\left( { - 1} \right) - 4{\left( { - 1} \right)^2} + 3 $ 

 $ p\left( { - 1} \right) =  - 5 - 4 + 3 $ 

 $ p\left( { - 1} \right) =  - 9 + 3 $ 

 $ p\left( { - 1} \right) =  - 6 $ 

Ans: The correct option is (D).

Here, we have  $ p\left( x \right) = x + 3 $ .

Now, on substituting  $ x $  with  $  - x $ , we get  $ p\left( { - x} \right) =  - x + 3 $ 

Now, on adding both the equations, we get

 $ p\left( x \right) + p\left( { - x} \right) = x + 3 - x + 3 $ 

 $ p\left( x \right) + p\left( { - x} \right) = 6 $ 

Ans: (C) is the correct answer.

Zero Polynomial is a constant Polynomial whose all coefficients are equal to 0. Zero of a Polynomial or we can say that root of a polynomial is the value of a variable that is responsible to make the Polynomial equals to zero. Hence, zero of the zero Polynomial is any real number.

Ans: (B) is the correct option.

To get the zero of the Polynomial, we have to put the expression equals to zero.

 $ p\left( x \right) = 0 $ 

 $ 2x + 5 = 0 $ 

 $ 2x =  - 5 $ 

 $ x =  - \dfrac{5}{2} $ 

Therefore,  $  - \dfrac{5}{2} $  is the zero of the Polynomial.

Ans: (B) is the correct option.

To find the zeroes of a Polynomial, we have to put that expression equals to zero.

Therefore,

 $ 2{x^2} + 7x - 4 = 0 $ 

Using middle term splitting,

 $ 2{x^2} + \left( {8 - 1} \right)x - 4 = 0 $ 

 $ 2{x^2} + 8x - x - 4 = 0 $ 

 $ 2x\left( {x + 4} \right) - 1\left( {x + 4} \right) = 0 $ 

 $ \left( {2x - 1} \right)\left( {x + 4} \right) = 0 $ 

So, there we have two Ansutions

 $ 2x - 1 = 0 $  and  $ x + 4 = 0 $ 

 $ 2x = 1 $  and  $ x =  - 4 $ 

 $ x = \dfrac{1}{2} $  and  $ x =  - 4 $