NCERT Solutions for Chapter 2 Polynomials Class 9 Maths

Chapter Name

NCERT Solutions for Chapter 2 Polynomials

Class

Class 9

Topics Covered

  • Algebraic Expressions
  • Polynomials
  • Remainder Theorem
  • Algebraic Identities

Related Study Materials

  • NCERT Solutions for Class 9 Maths
  • NCERT Solutions for Class 9
  • Revision Notes for Chapter 2 Polynomials Class 9 Maths
  • Important Questions for Chapter 2 Polynomials Class 9 Maths
  • MCQ for for Chapter 2 Polynomials Class 9 Maths

Short Revision for Ch 2 Polynomials Class 9 Maths 

  1. In the polynomial p(x), the highest exponent of x is called the degree of the polynomial p(x). 
  2. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 
  3. A linear polynomial in x is of the form ax + b, a ≠ 0. 
  4. A quadratic polynomial in x is of the form ax2 + bx + c, a ≠ 0. 
  5. A cubic polynomial in x is of the form ax3 + bx2 + cx + d,  a ≠ 0.
  6. A constant polynomial is free from a variable. 0, 7, 8, -4, etc. are examples of constant polynomials. 
  7. 0 is also called the zero polynomial. 
  8. The degree of a non - zero constant polynomial is 0. 
  9. The degree of the zero polynomial does not exist. 
  10. In the polynomial 3x3 - 4x + 7, the expressions 3x3 , - 4x and 7 are called the terms of the polynomial. 
  11. If a polynomial consists of only one variable, then the polynomial is called polynomial in one variable. 
  12. Polynomials consisting of one term, two terms and three terms are called monomial, binomial and trinomial respectively. 
  13. A real number k is a zero of the polynomial p(x), if p(k) = 0. 
  14. Every linear polynomial in one variable has a unique zero. 
  15. A non - zero constant polynomial has no zero. 
  16. Every real number is a zero of the zero polynomial. 
  17. A quadratic polynomial has at most two zeroes. 
  18. A cubic polynomial has at most three zeroes. 
  19. The polynomial equation of the polynomial p(x) is given by p(x) = 0. 
  20. 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). 
  21. Dividend = (Divisor × Quotient ) + Remainder. 
  22. 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). 
  23. Factor Theorem : If x -a is a factor of f(x). then f(a) = 0.
  24. (x + y)2 = x2 + 2xy + y2
  25. (x − y)2 = x2 − 2xy + y2
  26. x2 – y2 = (x + y)(x – y)
  27. (x + a)(a + b) = x2 + (a + b)x + ab
  28. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
  29. (x + y)3 = x3 + y3 + 3xy(x + y) = x3 + 3x2y + 3xy2 + y3
  30. (x – y)3 = x3 – y3 – 3xy(x – y) = x3 – 3x2y + 3xy2 – y3
  31. x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – zx)
  32. x3 + y3 + z3 = 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) 4x2 – 3x + 7
(ii) y2 + √2
(iii) 3√t + t√2
(iv) y + 2/y
(v) x10 + y3 + t50


2. Write the coefficients of x2 in each of the following:

(i) 2 + x+x
(ii) 2 - x+ x


(iv) √2x - 1


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) 5x3+4x2+7x
(ii) 4 - y

(iii) 5t - 
√7
(iv) 3

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) 7x3


Exercise 2.2 


1. Find the value of the polynomial (x)=5x−4x2+3 
(i) x = 0
(ii) x = – 1
(iii) x = 2


2. Find p(0), p(1) and p(2) for each of the following polynomials:
(i) p(y) = y− y + 1
(ii) p(t) = 2 + t + 2t2 – t3
(iii) p(x) = x3
(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 – π, x = 4/5
(iii) p(x) = x2 – 1, x = 1, −1
(iv) p(x) = (x + 1)(x – 2), x = -1, 2
(v) p(x) = x3 , x = 0
(vi) p(x) = lx + m, x = -m/l
(vii) p(x) = 3x2 – 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 


1. Find the remainder when x3+3x2+3x+1 is divided by
(i) x + 1
(ii) x -1/2 
(iii) x 
(iv) x + π
(v) 5 + 2x

2. Find the remainder when x3−ax2+6x−a is divided by x-a.

3. Check whether 7+3x is a factor of 3x3+7x.

Exercise 2.4 


1. Determine which of the following polynomials has (x + 1) a factor:
(i) x3+ x+ x + 1
(ii) x+ x+ x+ x + 1
(iii) x4+ 3x+ 3x+ x + 1
(iv) x– x2– (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) = 2x3+x2–2x–1, g(x) = x+1
(ii) p(x)=x3+3x2+3x+1, g(x) = x+2
(iii) p(x)=x3–4x2+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) = x2+x+k
(ii) p(x) = 2x2+kx+√2
(iii) p(x) = kx22x + 1
(iv) p(x) = kx– 3x + k


4. Factorize:
(i) 12x2–7x+1
(ii) 2x2+7x+3
(iii) 6x2+5x-6 
(iv) 3x2–x–4 


5. Factorize:
(i) x3–2x2–x +2
(ii) x3–3x2–9x–5
(iii) x3+13x2+32x+20
(iv) 2y3+y2–2y–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) (y2+3/2)(y2-3/2)
(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) 9x2+6xy+y2
(ii) 4y2−4y+1
(iii) x2–y2/100


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) 4x2+9y2+16z2+12xy–24yz–16xz
(ii ) 2x2+y2+8z2–2√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) 8a3+b3+12a2b+6ab2
(ii) 8a3–b3–12a2b+6ab2
(iii) 27–125a3–135a +225a2
(iv) 64a3–27b3–144a2b+108ab2
(v) 27p3–(1/216)−(9/2) p2+(1/4)p


9. Verify:
(i) x3+y= (x+y)(x2–xy+y2)
(ii) x3–y= (x–y)(x2+xy+y2)


10. Factorize each of the following:
(i) 27y3+125z3
(ii) 64m3–343n3

11. Factorise: 27x3+y3+z3– 9xyz. 


12. Verify that:
x3+y3+z3–3xyz = (1/2) (x+y+z)[(x–y)2+(y–z)2+(z–x)2]


13. If x+y+z = 0, show that x3+y3+z= 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 : 25a2–35a+12
(ii) Area : 35y2+13y–12


16. What are the possible expressions for the dimensions of the cuboids whose volumes are given below? 
(i) Volume : 3x– 12x
(ii) Volume : 12ky+ 8ky – 20k


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