NCERT Class 9 Solutions: Polynomials (Chapter 2) Exercise 2.1

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

  1. Equation

  2. Equation

  3. Equation

  4. Equation

  5. Equation

Solution:

Image explaining parts of a polynomial like variable, debree, coefficient

Parts of Polynomial

Image explaining parts of a polynomial like variable, debree, coefficient

A polynomial is an expression consisting of variables (or indeterminate) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. That is the exponent of the variable in a polynomial can only be zero or positive integer.

For example:

Equation is a polynomial but Equation or Equation are not.

Polynomials are defined as they are for a few distinct reasons: (1) because polynomials as functions have certain properties that other algebraic expressions don't have, and (2) because there are other terms for more generalized algebraic forms.

For example, polynomials are nice functions which don’t go to infinity (no poles). If we substitute a polynomial into another polynomial, we always get another polynomial. Also, all polynomials have a "degree" equal to the highest power of x in the polynomial, and a polynomial never has more roots (where value of polynomial becomes zero) than its degree.

  1. Equation

    There is only one variable x with whole number power

    So, this is a polynomial in one variable

  2. Equation

    There is only one variable y with whole number power

    So, this is a polynomial in one variable

  3. Equation

    There is one variable t but in Equation power of t is Equation which not a whole number

    So, Equation is not a polynomial.

  4. Equation

    There is only one variable y but Equation

    So, the power is not a whole number

    So, Equation is not a polynomial

  5. Equation

    There are three variables x, y and t

    And there powers are whole number

    So this polynomial in three variable

Q-2 Write the coefficients of Equation in each of the following:

  1. Equation

  2. Equation

  3. Equation

  4. Equation

Solution:

The numerical values (including +ve or –ve signs) of the terms in a polynomial are called the coefficients of the polynomial.

  1. Equation

    The coefficient of Equation is 1

  2. Equation

    The coefficient of Equation is -1

  3. Equation

    The coefficient of Equation is Equation

  4. Equation

    The coefficient of Equation is 0

Q-3 Given 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 a binomial of degree 35 can be written as Equation

  • Monomial has only one term in it. So monomial of degree 100 can be written as Equation

Q-4 Write the degree of each of the following polynomials:

  1. Equation

  2. Equation

  3. Equation

  4. Equation

Examples of polynomials, their degree number of terms and names

Examples of Polynomials

Examples of polynomials, their degree number of terms and names

Solution:

  1. Equation

    This is a polynomial in variable x and the highest power of variable x is 3

    Therefore, the degree of this polynomial is 3.

  2. Equation

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

    Therefore, the degree of this polynomial is 2.

  3. Equation

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

    Therefore, the degree of this polynomial is 1.

  4. Equation

    This is a constant polynomial

    Degree of a constant polynomial is always 0.

Q-5 Classify the following as linear, quadratic and cubic polynomials:

  1. Equation

  2. Equation

  3. Equation

  4. Equation

  5. Equation

  6. Equation

  7. Equation

Solution:

  1. The highest degree of Equation is 2

    So it is a quadratic polynomial.

  2. The highest degree of Equation is 3

    So, it is cubic polynomial.

  3. The highest degree of Equation is 2

    So it is a quadratic polynomial.

  4. The highest degree of Equation is 1

    So it is a linear polynomial.

  5. The highest degree of Equation is 1

    So it is a linear polynomial.

  6. The highest degree of Equation is 2

    So it is a quadratic polynomial.

  7. The highest degree of Equation is 3

    So, it is cubic polynomial.

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