Class- XI-CBSE-Mathematics Polynomials
CBSE NCERT Solutions for Class 9 Mathematics Chapter 2
Back of Chapter Questions
Exercise: 2.1
1. Which of the following expressions are polynomials in one variable and which are
not? State reasons for your answer.
(i) 4𝑥𝑥 2 − 3x + 7
(ii) y 2 + √2
(iii) 3√t + t√2
2
(iv) y+y
(v) 𝑥𝑥10 + y 3 + t 50
Solution:
(i) Given expression is a polynomial
It is of the form an 𝑥𝑥 𝑛𝑛 + an−1 𝑥𝑥 n−1 + ⋯ + a1 𝑥𝑥 + a0 where an , an−1 , … a0
are constants. Hence given expression 4x 2 − 3x + 7 is a polynomial.
(ii) Given expression is a polynomial
It is of the form an x n + an−1 x n−1 + ⋯ + a1 x + a0 where an , an−1 , … a0
are constants. Hence given expression y 2 + √2 is a polynomial.
(iii) Given expression is not a polynomial. It is not in the form of
an x n + an−1 2n−1 + ⋯ + a1 x + a0
where an , an−1 , … a0 all constants.
Hence given expression 3√t + t√2 is not a polynomial.
(iv) Given expression is not a polynomial
2
y+ = y + 2. y −1
y
It is not of form an x n + an−1 x n−1 + ⋯ + a0 , where an , an−1 , … a0 are
constants.
2
Hence given expression y + y is not a polynomial.
(v) Given expression is a polynomial in three variables. It has three variables
x, y, t.
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Hence the given expression x10 + y 3 + t 50 is not a polynomial in one
variable.
2. Write the coefficients of x 2 in each of the following:
(i) 2 + x2 + x
(ii) 2 − x2 + x3
π 2
(iii) 2
x +x
(iv) √2x − 1
Solution:
(i) The constant multiplied with the term x 2 is called the coefficient of the x 2 .
Given polynomial is 2 + x 2 + x.
Hence, the coefficient of x 2 in given polynomial is equal to 1.
(ii) The constant multiplied with the term x 2 is called the coefficient of the x 2 .
Given polynomial is 2 − x 2 + x 3 .
Hence, the coefficient of x 2 in given polynomial is equal to −1.
(iii) The constant multiplied with the term x 2 is called the coefficient of the x 2 .
π
Given polynomial is 2 x 2 + x.
π
Hence, the coefficient of x 2 in given polynomial is equal to 2 .
(iv) The constant multiplied with the term x 2 is called the coefficient of the x 2 .
Given polynomial is √2x − 1.
In the given polynomial, there is no x 2 term.
Hence, the coefficient of x 2 in given polynomial is equal to 0.
3. Give one example each of a binomial of degree 35 and of a monomial of degree
100o .
Solution:
Degree of polynomial is highest power of variable in the polynomial. And number
of terms in monomial and binomial respectively equals to one and two.
A binomial of degree 35 can be x 35 + 7
A monomial of degree 100 can be 2x100 + 9
4. Write the degree of each of the following polynomials
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,Class- XI-CBSE-Mathematics Polynomials
(i) 5x 3 + 4x 2 + 7x
(ii) 4 − y2
(iii) 5𝑡𝑡 − √7
(iv) 3
Solution:
(i) Degree of polynomial is highest power of variable in the polynomial.
Given polynomial is 5x 3 + 4x 2 + 7x
Hence, the degree of given polynomial is equal to 3.
(ii) Degree of polynomial is highest power of variable in the polynomial.
Given polynomial is 4 − y 2
Hence, the degree of given polynomial is 2.
(iii) Degree of polynomial is highest power of variable in the polynomial
Given polynomial is 5t − √7
Hence, the degree of given polynomial is 1.
(iv) Degree of polynomial 1, highest power of variable in the polynomial.
Given polynomial is 3.
Hence, the degree of given polynomial is 0.
5. Classify the following as linear, quadratic and cubic polynomials.
(i) x2 + x
(ii) x − x3
(iii) y + y2 + 4
(iv) 1+x
(v) 3t
(vi) r2
(vii) 7x 3
Solution:
(i) Linear, quadratic, cubic polynomials have degrees 1, 2, 3 respectively.
Given polynomial is x 2 + x
It is a quadratic polynomial as its degree is 2.
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, Class- XI-CBSE-Mathematics Polynomials
(ii) Linear, quadratic, cubic polynomials have its degree 1, 2, 3 respectively.
Given polynomial is x − x 3 .
It is a cubic polynomial as its degree is 3.
(iii) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is y + y 2 + 4.
It is a quadratic polynomial as its degree is 2.
(v) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is 1 + x.
It is a linear polynomial as its degree is 1.
(v) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is 3t
It is a linear polynomial as its degree is 1.
(vi) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is r 2 .
It is a quadratic polynomial as its degree is 2.
(vii) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is 7x 3 .
It is a cubic polynomial as its degree is 3.
Exercise: 2.2
1. Find the value of the polynomial 5x − 4x 2 + 3 at
(i) x=0
(ii) x = −1
(iii) x=2
Solution:
(i) Given polynomial is 5x − 4x 2 + 3
Value of polynomial at x = 0 is 5(0) − 4(0)2 + 3
=0−0+3
=3
Therefore, value of polynomial 5x − 4x 2 + 3 at x = 0 is equal to 3.
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CBSE NCERT Solutions for Class 9 Mathematics Chapter 2
Back of Chapter Questions
Exercise: 2.1
1. Which of the following expressions are polynomials in one variable and which are
not? State reasons for your answer.
(i) 4𝑥𝑥 2 − 3x + 7
(ii) y 2 + √2
(iii) 3√t + t√2
2
(iv) y+y
(v) 𝑥𝑥10 + y 3 + t 50
Solution:
(i) Given expression is a polynomial
It is of the form an 𝑥𝑥 𝑛𝑛 + an−1 𝑥𝑥 n−1 + ⋯ + a1 𝑥𝑥 + a0 where an , an−1 , … a0
are constants. Hence given expression 4x 2 − 3x + 7 is a polynomial.
(ii) Given expression is a polynomial
It is of the form an x n + an−1 x n−1 + ⋯ + a1 x + a0 where an , an−1 , … a0
are constants. Hence given expression y 2 + √2 is a polynomial.
(iii) Given expression is not a polynomial. It is not in the form of
an x n + an−1 2n−1 + ⋯ + a1 x + a0
where an , an−1 , … a0 all constants.
Hence given expression 3√t + t√2 is not a polynomial.
(iv) Given expression is not a polynomial
2
y+ = y + 2. y −1
y
It is not of form an x n + an−1 x n−1 + ⋯ + a0 , where an , an−1 , … a0 are
constants.
2
Hence given expression y + y is not a polynomial.
(v) Given expression is a polynomial in three variables. It has three variables
x, y, t.
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,Class- XI-CBSE-Mathematics Polynomials
Hence the given expression x10 + y 3 + t 50 is not a polynomial in one
variable.
2. Write the coefficients of x 2 in each of the following:
(i) 2 + x2 + x
(ii) 2 − x2 + x3
π 2
(iii) 2
x +x
(iv) √2x − 1
Solution:
(i) The constant multiplied with the term x 2 is called the coefficient of the x 2 .
Given polynomial is 2 + x 2 + x.
Hence, the coefficient of x 2 in given polynomial is equal to 1.
(ii) The constant multiplied with the term x 2 is called the coefficient of the x 2 .
Given polynomial is 2 − x 2 + x 3 .
Hence, the coefficient of x 2 in given polynomial is equal to −1.
(iii) The constant multiplied with the term x 2 is called the coefficient of the x 2 .
π
Given polynomial is 2 x 2 + x.
π
Hence, the coefficient of x 2 in given polynomial is equal to 2 .
(iv) The constant multiplied with the term x 2 is called the coefficient of the x 2 .
Given polynomial is √2x − 1.
In the given polynomial, there is no x 2 term.
Hence, the coefficient of x 2 in given polynomial is equal to 0.
3. Give one example each of a binomial of degree 35 and of a monomial of degree
100o .
Solution:
Degree of polynomial is highest power of variable in the polynomial. And number
of terms in monomial and binomial respectively equals to one and two.
A binomial of degree 35 can be x 35 + 7
A monomial of degree 100 can be 2x100 + 9
4. Write the degree of each of the following polynomials
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,Class- XI-CBSE-Mathematics Polynomials
(i) 5x 3 + 4x 2 + 7x
(ii) 4 − y2
(iii) 5𝑡𝑡 − √7
(iv) 3
Solution:
(i) Degree of polynomial is highest power of variable in the polynomial.
Given polynomial is 5x 3 + 4x 2 + 7x
Hence, the degree of given polynomial is equal to 3.
(ii) Degree of polynomial is highest power of variable in the polynomial.
Given polynomial is 4 − y 2
Hence, the degree of given polynomial is 2.
(iii) Degree of polynomial is highest power of variable in the polynomial
Given polynomial is 5t − √7
Hence, the degree of given polynomial is 1.
(iv) Degree of polynomial 1, highest power of variable in the polynomial.
Given polynomial is 3.
Hence, the degree of given polynomial is 0.
5. Classify the following as linear, quadratic and cubic polynomials.
(i) x2 + x
(ii) x − x3
(iii) y + y2 + 4
(iv) 1+x
(v) 3t
(vi) r2
(vii) 7x 3
Solution:
(i) Linear, quadratic, cubic polynomials have degrees 1, 2, 3 respectively.
Given polynomial is x 2 + x
It is a quadratic polynomial as its degree is 2.
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, Class- XI-CBSE-Mathematics Polynomials
(ii) Linear, quadratic, cubic polynomials have its degree 1, 2, 3 respectively.
Given polynomial is x − x 3 .
It is a cubic polynomial as its degree is 3.
(iii) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is y + y 2 + 4.
It is a quadratic polynomial as its degree is 2.
(v) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is 1 + x.
It is a linear polynomial as its degree is 1.
(v) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is 3t
It is a linear polynomial as its degree is 1.
(vi) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is r 2 .
It is a quadratic polynomial as its degree is 2.
(vii) Linear, quadratic, cubic polynomial has its degree 1, 2, 3 respectively.
Given polynomial is 7x 3 .
It is a cubic polynomial as its degree is 3.
Exercise: 2.2
1. Find the value of the polynomial 5x − 4x 2 + 3 at
(i) x=0
(ii) x = −1
(iii) x=2
Solution:
(i) Given polynomial is 5x − 4x 2 + 3
Value of polynomial at x = 0 is 5(0) − 4(0)2 + 3
=0−0+3
=3
Therefore, value of polynomial 5x − 4x 2 + 3 at x = 0 is equal to 3.
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