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. 4x23x+7

  2. y2+2

  3. 3t+t2

  4. y+2y

  5. x10+y3+t50

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:

1+2x+2x2 is a polynomial but 12x or x 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. 4x23x+7

    There is only one variable x with whole number power

    So, this is a polynomial in one variable

  2. y2+2

    There is only one variable y with whole number power

    So, this is a polynomial in one variable

  3. 3t+t2

    There is one variable t but in 3t power of t is 12 which not a whole number

    So, 3t+t2 is not a polynomial.

  4. y+2y

    There is only one variable y but 2y=2y1

    So, the power is not a whole number

    So, y+2y is not a polynomial

  5. x10+y3+t50

    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 x2 in each of the following:

  1. 2+x2+x

  2. 2x2+x3

  3. π2x2+x

  4. 2x1

Solution:

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

  1. 2+x2+x

    The coefficient of x2 is 1

  2. 2x2+x3

    The coefficient of x2 is -1

  3. π2x2+x

    The coefficient of x2 is π2

  4. 2x1=0×x2+2x1

    The coefficient of x2 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 x35+1

  • Monomial has only one term in it. So monomial of degree 100 can be written as x100

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

  1. 5x3+4x2+7x

  2. 4y2

  3. 5t7

  4. 3

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. 5x3+4x2+7x

    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. 4y2

    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. 5t7

    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. 3

    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. x2+x

  2. xx3

  3. y+y2+4

  4. 1+x

  5. 3t

  6. r2

  7. 7x3

Solution:

  1. The highest degree of x2+x is 2

    So it is a quadratic polynomial.

  2. The highest degree of xx3 is 3

    So, it is cubic polynomial.

  3. The highest degree of y+y2+4 is 2

    So it is a quadratic polynomial.

  4. The highest degree of 1+x is 1

    So it is a linear polynomial.

  5. The highest degree of 3t is 1

    So it is a linear polynomial.

  6. The highest degree of r2 is 2

    So it is a quadratic polynomial.

  7. The highest degree of 7x3 is 3

    So, it is cubic polynomial.

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