NCERT Class 9 Maths Solutions Chapter - 2 Polynomials, Ex 2.1

Ex - 2.1

Question 1.  Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer


Solution

(i)     

         Yes, this expression is a polynomial in one variable x.

(ii)     

          Yes, this expression is a polynomial in one variable y.

(iii)     

           No, here the exponent of variable t in term  which is not a whole number.

           So this expression is not a polynomial.

(iv)     

          No, here the exponent of variable t in term  is -1, which is not a whole number. So this expression is not a polynomial.

(v)     

          No, this expression is a polynomial in 3 variables x, y and t.

Concept Insight: In such problems to check whether the given algebraic expressions is a polynomial or not, check the exponents of variable to be a whole number. The second step is to look for the number of variables the expression has. Any alphabet used in the expression is the variable unless specified as constant.

Question 2.  Write the coefficient of x² in each of the following:

Solution

(i)     

         In the above expression coefficient of x² is 1

(ii)     

          In the above expression coefficient of x² is - 1.

(iii)     

           In the above expression coefficient of  

(iv)     

            In the above expression coefficient of x²  is 0.

Concept Insight: The constant/ variable multiplied with the variable is the coefficient of the variable. Also consider the sign (+ve or the -ve) of the term while writing the coefficient.

Question 3.  Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Solution

Degree of a polynomial is the highest power of variable in the polynomial.Binomial has two terms in it. So binomial of degree 35 can be written as x³⁵ + 7 .   Monomial has only one term in it. So monomial of degree 100 can be written as 7x¹⁰⁰.   Concept Insight: Mono, bi and tri means one, two and three respectively. So, monomial is a polynomial having one term similarly for binomials and trinomials. Degree is the highest exponent of variable.  The answer is not unique in such problems . Remember that the terms are always separated by +ve or -ve sign and not with.

Question 4.  Write the degree of each of the following polynomials:

Solution

 Degree of a polynomial is the highest power of variable in the polynomial.

          This is a polynomial in variable x and highest power of variable x is 3. So, degree of

          this polynomial is 3

           This is a polynomial in variable y and the highest power of variable y is 2. So degree

           of this polynomial is 2.

           This is a polynomial in variable t and the highest power of variable t is 1. So degree

           of this polynomial is 1.

(iv)      3

           This is a constant polynomial. Degree of a constant polynomial is always 0.

Concept Insight: Degree is the highest  exponent of the variable. While finding the degree of a polynomial express the polynomial in standard form  Combine the like terms and Remember the result that x^o = 1.

Question 5.  Classify the following as linear, quadratic and cubic polynomial:

Solution

(i)       2 + x² + x  is a quadratic polynomial as its degree is 2.

(ii)      x - x³ is a cubic polynomial as  its degree is 3.

(iii)     y + y² + 4  is a quadratic polynomial as its degree is 2.

(iv)     1 + x is a linear polynomial as its degree is 1.

(v)      3t is a linear polynomial as its degree is 1.

(vi)     r² is a quadratic polynomial as its degree is 2.

(vii)     7x³ is a cubic polynomial as its degree is 3.

Concept Insight: Linear polynomial, quadratic polynomial and cubic polynomial has its degrees as 1, 2, and 3 respectively.

Ex-2.2