NCERT Solutions for Chapter 2 Polynomials Class 9 Maths
Chapter Name  NCERT Solutions for Chapter 2 Polynomials 
Class  Class 9 
Topics Covered 

Related Study Materials 

Short Revision for Ch 2 Polynomials Class 9 Maths
 In the polynomial p(x), the highest exponent of x is called the degree of the polynomial p(x).
 Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
 A linear polynomial in x is of the form ax + b, a ≠ 0.
 A quadratic polynomial in x is of the form ax^{2} + bx + c, a ≠ 0.
 A cubic polynomial in x is of the form ax^{3} + bx^{2} + cx + d, a ≠ 0.
 A constant polynomial is free from a variable. 0, 7, 8, 4, etc. are examples of constant polynomials.
 0 is also called the zero polynomial.
 The degree of a non  zero constant polynomial is 0.
 The degree of the zero polynomial does not exist.
 In the polynomial 3x^{3}  4x + 7, the expressions 3x^{3} ,  4x and 7 are called the terms of the polynomial.
 If a polynomial consists of only one variable, then the polynomial is called polynomial in one variable.
 Polynomials consisting of one term, two terms and three terms are called monomial, binomial and trinomial respectively.
 A real number k is a zero of the polynomial p(x), if p(k) = 0.
 Every linear polynomial in one variable has a unique zero.
 A non  zero constant polynomial has no zero.
 Every real number is a zero of the zero polynomial.
 A quadratic polynomial has at most two zeroes.
 A cubic polynomial has at most three zeroes.
 The polynomial equation of the polynomial p(x) is given by p(x) = 0.
 Remainder Theorem : If p(x) is a polynomial of degree greater than or equal to 1 and f(x) is divided by the linear polynomial x  a, then the remainder is f(a).
 Dividend = (Divisor × Quotient ) + Remainder.
 If dividend, divisor, quotient and remainder are respectively f(x), g(x), q(x) and r(x) such that degree of f(x) ≥ degree of g(x) with g(x) ≠ 0, then f(x) = g(x) × q(x) + r(x) where r(x) = 0 or degree of r(x) < degree of g(x).
 Factor Theorem : If x a is a factor of f(x). then f(a) = 0.
 (x + y)^{2} = x^{2} + 2xy + y^{2}
 (x − y)^{2} = x^{2} − 2xy + y^{2}
 x^{2} – y^{2} = (x + y)(x – y)
 (x + a)(a + b) = x^{2} + (a + b)x + ab
 (x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx
 (x + y)^{3} = x^{3} + y^{3} + 3xy(x + y) = x^{3} + 3x^{2}y + 3xy^{2} + y^{3}
 (x – y)^{3} = x^{3} – y^{3} – 3xy(x – y) = x^{3} – 3x^{2}y + 3xy^{2} – y^{3}
 x^{3} + y^{3} + z^{3} – 3xyz = (x + y + z)(x^{2} + y^{2} + z^{2} – xy – yz – zx)
 x^{3} + y^{3} + z^{3} = 3xyz, if x + y + x = 0
Exercise 2.1
1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x^{2} – 3x + 7
(ii) y^{2} + √2
(iii) 3√t + t√2
(iv) y + 2/y
(v) x^{10} + y^{3} + t^{50}
(i) 2 + x^{2 }+x
(ii) 2  x^{2 }+ x^{3 }
3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
4. Write the degree of each of the following polynomials:
(i) 5x^{3}+4x^{2}+7x
(ii) 4  y^{2 }
(iii) 5t  √7
(iv) 3
5. Classify the following as linear, quadratic and cubic polynomials:
(i) x^{2} + x
(ii) x – x^{3}(iii) y + y^{2} + 4
(iv) 1 + x
(v) 3t
(vi) r^{2}(vii) 7x^{3}
Exercise 2.2
(i) x = 0
(ii) x = – 1
(iii) x = 2
(i) p(y) = y^{2 }− y + 1
(ii) p(t) = 2 + t + 2t^{2} – t^{3}(iii) p(x) = x^{3}(iv) p(x) = (x – 1)(x+ 1)
3. Verify whether the following are zeroes of the polynomial, indicated against them.
(i) p(x) = 3x +1, x = −1/3
(ii) p(x) = 5x – Ï€, x = 4/5
(iii) p(x) = x^{2} – 1, x = 1, −1
(iv) p(x) = (x + 1)(x – 2), x = 1, 2
(v) p(x) = x^{3} , x = 0
(vi) p(x) = lx + m, x = m/l
(vii) p(x) = 3x^{2} – 1, x = 1/√3, 2/√3
(viii) p(x) = 2x + 1, x = 1/2
4. Find the zero of the polynomials in each of the following cases:
(i) p(x) = x+5
(ii) p(x) = x  5
(iii) p(x) = 2x + 5
(iv) p(x) = 3x  2
(v) p(x) = 3x
(vi) p(x) = ax, a ≠ 0
(vii) p(x) = cx + d, c ≠ 0, c, d are real numbers.
Exercise 2.3
(i) x + 1
(ii) x 1/2
(iii) x
(iv) x + Ï€
(v) 5 + 2x
Exercise 2.4
1. Determine which of the following polynomials has (x + 1) a factor:
(i) x^{3}+ x^{2 }+ x + 1
(ii) x^{4 }+ x^{3 }+ x^{2 }+ x + 1
(iii) x^{4}+ 3x^{3 }+ 3x^{2 }+ x + 1
(iv) x^{3 }– x^{2}– (2+√2)x +√2
2. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x) = 2x^{3}+x^{2}–2x–1, g(x) = x+1
(ii) p(x)=x^{3}+3x^{2}+3x+1, g(x) = x+2
(iii) p(x)=x^{3}–4x^{2}+x+6, g(x) = x–3
3. Find the value of k, if x–1 is a factor of p(x) in each of the following cases:
(i) p(x) = x^{2}+x+k
(ii) p(x) = 2x^{2}+kx+√2
(iii) p(x) = kx^{2}–√2x + 1
(iv) p(x) = kx^{2 }– 3x + k
4. Factorize:
(i) 12x^{2}–7x+1
(ii) 2x^{2}+7x+3
(iii) 6x^{2}+5x6
(iv) 3x^{2}–x–4
5. Factorize:
(i) x^{3}–2x^{2}–x +2
(ii) x^{3}–3x^{2}–9x–5
(iii) x^{3}+13x^{2}+32x+20
(iv) 2y^{3}+y^{2}–2y–1
Exercise 2.5
1. Use suitable identities to find the following products:
(i) (x+4)(x +10)
(ii) (x+8)(x –10)
(iii) (3x+4)(3x–5)
(iv) (y^{2}+3/2)(y^{2}3/2)
(v) (3  2x)(3 + 2x).
2. Evaluate the following products without multiplying directly:
(i) 103×107
(ii) 95×96
(iii) 104×96
3. Factorize the following using appropriate identities:
(i) 9x^{2}+6xy+y^{2}(ii) 4y^{2}−4y+1
(iii) x^{2}–y^{2}/100
4. Expand each of the following, using suitable identities:
(i) (x+2y+4z)^{2}(ii) (2x−y+z)^{2}(iii) (−2x+3y+2z)^{2}(iv) (3a –7b–c)^{2}(v) (–2x+5y–3z)^{2}(vi) (1/4)a(1/2)b +1)^{2}
5. Factorize:
(i) 4x^{2}+9y^{2}+16z^{2}+12xy–24yz–16xz
(ii ) 2x^{2}+y^{2}+8z^{2}–2√2xy+4√2yz–8xz
6. Write the following cubes in expanded form:
(i) (2x+1)^{3}(ii) (2a−3b)^{3}(iii) ((3/2)x+1)^{3}(iv) (x−(2/3)y)^{3}
7. Evaluate the following using suitable identities:
(i) (99)^{3}(ii) (102)^{3}(iii) (998)^{3}
8. Factorise each of the following:
(i) 8a^{3}+b^{3}+12a^{2}b+6ab^{2}(ii) 8a^{3}–b^{3}–12a^{2}b+6ab^{2}(iii) 27–125a^{3}–135a +225a^{2}(iv) 64a^{3}–27b^{3}–144a^{2}b+108ab^{2}(v) 27p^{3}–(1/216)−(9/2) p^{2}+(1/4)p
9. Verify:
(i) x^{3}+y^{3 }= (x+y)(x^{2}–xy+y^{2})
(ii) x^{3}–y^{3 }= (x–y)(x^{2}+xy+y^{2})
10. Factorize each of the following:
(i) 27y^{3}+125z^{3}(ii) 64m^{3}–343n^{3}
11. Factorise: 27x^{3}+y^{3}+z^{3}– 9xyz.
12. Verify that:
x^{3}+y^{3}+z^{3}–3xyz = (1/2) (x+y+z)[(x–y)^{2}+(y–z)^{2}+(z–x)^{2}]
13. If x+y+z = 0, show that x^{3}+y^{3}+z^{3 }= 3xyz.
14. Without actually calculating the cubes, find the value of each of the following:
(i) (−12)^{3}+(7)^{3}+(5)^{3}(ii) (28)^{3}+(−15)^{3}+(−13)^{3}
15. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:
(i) Area : 25a^{2}–35a+12
(ii) Area : 35y^{2}+13y–12
16. What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
(i) Volume : 3x^{2 }– 12x
(ii) Volume : 12ky^{2 }+ 8ky – 20k