A polynomial is a finite expression constructed from variables and constants. using the operations of addition. subtraction. multiplication and taking non-negative integer powers it can be written as the sum of a finite number of terms.
POLYNOMIALS :-a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power (such as a + bx + cx2)
Types of Polynomials
Polynomials are categorised into three groups depending upon the number of terms it comprises of. Here are the types of polynomials.
Monomial
A monomial is a type of polynomial in algebra consisting of a single non-zero term. A polynomial expression consists of one or more terms. Therefore, every term of a polynomial expression is a monomial. Every numeric value such as 6, 12, 151 is a monomial by itself, whereas the variables can x, y, a can also be included in the list of monomials in algebra. Example of a monomial expression – 7x².
Rules for monomial algebraic expressions:
If a monomial is multiplied by a constant, the output will also be a monomial.
If a monomial is multiplied by a monomial, the result will also be a monomial. For instance, if a monomial three is multiplied by 3, the result 8 is also a monomial.
Binomial
A binomial is a type of polynomial expression comprising of two non-zero terms. Let’s see some examples to make it clear,
7x² + 8y is a binomial expression with two variables.
10x⁴ + 9y is also a binomial expression with two variables.
Trinomial
A trinomial is a type of polynomial expression comprising of three non-zero terms. Let’s see some examples to make it clear,
5x²+8x+9 is a trinomial expression with one variables x.
a + b+ c is a trinomial expression with three variables.
7x – 6y + 9z is a trinomial expression with three variables.
Polynomial Theorems
Some of the vital theorems of polynomials are as follows:
Remainder Theorem
The polynomial remainder theorem, also quoted as the little Bezout’s theorem, implies that if a polynomial P(x) is divided by any linear polynomial depicted by (x – a), the remainder of the operation will be a constant given by P(a), i.e., r = P(a).
Factor Theorem
The factor theorem implies that if P(x) is a polynomial of degree n > 1, and ‘a’ is a real number, this portrays that:
If P(x) = 0, then (x – a) is the factor of P(x),
If (x – a) is the factor of P(x), P(x) = 0.
Bezout’s Theorem
Bezout’s Theorem states that if P(x) = 0, then P(x) gets divided by (x – a), with ‘r’ as the remainder.
Intermediate Value Theorem
The intermediate value theorem states that when a polynomial function transforms from a negative to a positive value, it must cross the x-axis. In other words, the theorem highlights the properties of continuity of a function.
Fundamental Theorem of Algebra
The fundamental theorem of algebra states that each non-constant single variable that consists of a complex coefficient possess a minimum of one complex root.
Degree of a Polynomial : The highest power of x in a polynomial p(x) is called the degree of the polynomial.
Example 1. Write the degree of the polynomial:
x2+2x+4. The degree of the equation is 2 . i.e. the highest power of variable in the equation.
Degree of a Polynomial: If p(x) is a polynomial in x and ‘a’ is a real number, then the value obtained by putting x = a in p(x), is the value of p(x) at x = a and is denoted by p(a).
Example 2. Find the value of the polynomial
p(x) = 5x – 4x² + 3
On putting x = 0 ,
p(0) = 5(0) – 4(0)² + 3
= 0 – 0 + 3
=3
p(0) = 3
constant polynomial : A polynomial having its highest degree zero is called a constant polynomial. It has no variables, only constants.
For example: f(x) = 12, g(x) = -51 , h(y) = 3/2 etc are constant polynomials.
linear polynomial : A polynomial having its highest degree one is called a linear polynomial. For example, f(x) = x- 2, g(x) = 2 x , h(x) = -9x + 8 are linear polynomials. In general g(x) = ax + b , a ≠ 0 is a linear polynomial.
Quadratic polynomial :- A quadratic polynomial is a polynomial of degree 2. A univariate quadratic polynomial has the form. . An equation involving a quadratic polynomial is called a quadratic equation.
9th Class Formulas For Algebraic Identities
Given below are the algebraic identities which are considered very important maths formulas for Class 9.
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- (a + b) (a – b) = a2 -b2
- (x + a) (x + b) = x2 + (a + b) x + ab
- (x + a) (x – b) = x2 + (a – b) x – ab
- (x – a) (x + b) = x2 + (b – a) x – ab
- (x – a) (x – b) = x2 – (a + b) x + ab
- (a + b)3 = a3 + b3 + 3ab (a + b)
- (a – b)3 = a3 – b3 – 3ab (a – b)
- (x + y + z)2 = x2 + y2 + z2 + 2xy +2yz + 2xz
- (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
- (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
- (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
- x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz -xz)
- x2 + y2 =
[(x + y)2 + (x – y)2]
- (x + a) (x + b) (x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc
- x3 + y3 = (x + y) (x2 – xy + y2)
- x3 – y3 = (x – y) (x2 + xy + y2)
- x2 + y2 + z2 – xy – yz – zx =
MCQs on Class 9 Maths Chapter 2 Polynomials
Check the multiple-choice questions for 9th Class Maths Polynomial chapter. Each MCQ will have four options here, out of which only one is correct. Students have to pick the correct option and check the answer provided here.
Download this file of Polynomial Extra Question for practice
(a) x–2 + x–1 + 5
(b)
(c)
(d) 3x2 + 7
Answer:- d
is
Q16. x2-2x+1 is a polynomial in:
a. One Variable
b. Two Variables
c. Three variable
d. None of the above
Answer: a
Explanation: x2-2x+1 can be written as x2-2x1+1x0. Hence, we can see
that x is the only variable having powers as whole numbers: 2,1 and 0.
Q17. The coefficient of x2 in 3x3+2x2-x+1 is:
a. 1
b. 2
c. 3
d. -1
Answer: b
Explanation: The coefficient of x2 in equation 3x3+2x2-x+1 is the
multiple of x2.
Q18. A binomial of degree 20 in the following is:
a. 20x + 1
b. x/20 + 1
c. x20 +1
d. x2+20
Answer: c
Explanation: A polynomial having two terms and the highest degree 20
is called a binomial of degree 20.
Q19. The degree of 4x3-12x2+3x+9 is
a. 0
b. 1
c. 2
d. 3
Answer: d
Explanation: The degree is the highest power of a variable in an equation.
Q20. x2 – x is ________ polynomial.
a. Linear
b. Quadratic
c. Cubic
d. None of the above
Answer: b
Explanation: A polynomial of degree two is known as a quadratic polynomial.
Q21. x – x3 is a ________ polynomial.
a. Linear
b. Quadratic
c. Cubic
d. None of the above
Answer: c
Explanation: A polynomial of degree three is known as a cubic polynomial.
Q22. 1+3x is a _________ polynomial.
a. Linear
b. Quadratic
c. Cubic
d. None of the above
Answer: a
Explanation: A polynomial of degree one is known as a linear polynomial.
Q23. The value of f(x) = 5x−4x2+3 when x = -1, is:
a. 3
b. -12
c. -6
d. 6
Answer: c
Explanation: When x= -1
f(x)=5x−4x2+3
f(−1)=5(−1) −4(−1)2+3
=−5–4+3
=−6
Q24. The value of p(t) = 2+t+2t2−t3 when t=0 is
a. 2
b. 1
c. 4
d. 0
Answer: a
Explanation: p(0)=2+0+2(0)2–(0)3=2
Q25. The zero of the polynomial f(x) = 2x+7 is
a.
b. –
c. 7/2
d. -7/2
Answer: d
Explanation: f(x)=2x+7
⇒2x+7=0
⇒2x=−7
⇒x=−7/2
∴x = −7/2 is a zero polynomial of the polynomial f(x).
Q26. What is the degree of the polynomial √3?
a. 0
b. 1
c. 1/2
d. 2
Answer: c
Explanation: The polynomial √3 can also be written as 31/2.
Hence, the degree of the polynomial √3 is 1/2.
Q27. The degree of the constant polynomial is
a. 0
b. 1
c. 2
d. 3
Answer: a
Explanation: The degree of the constant polynomial is 0. For example, 3 is a constant polynomial that is equal to 3x0, and its degree is 0.
Q28. One of the linear factors of 3x2+8x+5 is
a. (x+1)
b. (x-2)
c. (x+2)
d. (x-4)
Answer: a
Explanation: 3x2+8x+5 = 3x2+ 3x+5x+5
3x2+8x+5 = 3x(x+1)+5(x+1)
3x2+8x+5 = (3x+5)(x+1)
Therefore, (x+1) is one of the factors of 3x2+8x+5.
Q29. The coefficient of x in 7x2+6x-2 is
a. 2
b. 6
c. -2
d. 7
Answer: b
Explanation: The coefficient of x in 7x2+6x-2 is 6. Because the number
multiplied by x is 6.
Q30. Which of the following is an example of the quadratic polynomial?
a. 7x+3
b. 2x2+x-1
c. x+3x3-9
d. None of the above
Answer: b
Explanation: 2x2+x-1 is a quadratic polynomial because the highest
degree of the polynomial is 2.
Q31. Find the value of 72-52.
a. 22
b. 23
c. 24
d. 25
Answer: c
Explanation: 72-52 = 49 – 25 = 24.
Q32. If x2+kx+6 = (x+2)(x+3) for all k, find the value of k.
a. -1
b. 1
c. 3
d. 5
Answer: d
Explanation: x2+kx+6 = (x+2)(x+3)
x2+kx+6 = x2+3x+2x+6
x2+kx+6 = x2+5x+6
Hence, the value of k is 5.
Q33. What is the zero of the polynomial p(x)=cx+d?
a. -c
b. -d
c. -d/c
d. d/c
Answer: c
Explanation: The zero of the polynomial p(x)= cx+d is -d/c.
cx+d = 0
cx = -d
x = -d/c.
Q34. The zero of the polynomial p(x) = -5x+5 is
a. 0
b. -5
c. -1
d. 1
Answer: d
Explanation: p(x) = -5x+5
-5x+5 = 0
-5x = -5
x = -5/-5 =1
Q35. which of the following is a constant polynomial?
a. 4x+1
b. 3
c. 2x2
d. 6x+3
Answer: b
Explanation: 3 is a constant polynomial, as 3 = 3x0. Whereas 4x+1 and 6x+3 are linear polynomial and 2x2 is a quadratic polynomial.
Q36. Find the value of 525² – 475².
(a) 100
(b) 10000
(c) 50000
(d) 100000
Answer: c
Q37. One of the factors of (1 + 3y)² + (9y² – 1) is
(a) 1 – 3y
(b) 3 – y
(c) 3y + 1
(d) y – 3
Answer: c
Q38. One of the factors of (x³ – 1) – (x – 1) is
(a) x² + 1
(b) x² – 1
(c) x – 1
(d) x + 4
Answer: c
Q39. Find the value of k if x² + kx + 6 = (x + 2) (x + 3) for all k.
(a) 1
(b) -1
(c) 5
(d) 3
Answer: c
Q40. If x – 2 is a factor of 5x² – kx – 18, then find the value of k.
(a) -1
(b) 1
(c) 0
(d) 5
Answer: b
Q41. Find the coefficient of x² in (3x² – 5) (4 + 4x²).
(a) 12
(b) 5
(c) -8
(d) 8
Answer: c
Q42. Find the value of p for which x + p is a factor of x² + px + 3 – p.
(a) 1
(b) -1
(c) 3
(d) -3
Answer: c
Q43. Find the remainder on dividing 5y³ – 2y² – 7y + 1 by y.
(a) -1
(b) 1
(c) 0
(d) 2
Answer: b
Q44. One of the factors of (16y² – 1) + (1 – 4y)²
(a) (4 + a)
(b) (4 – y)
(c) (4y + 1)
(d) 8y
Answer: d
Q45. If a + b + c = 0, then the value of a³ + b³ + c³
(a) abc
(b) -3abc
(c) 0
(d) 3abc
Answer: d
Q46. If the area of the rectangle is 4x² + 4x – 3, then find its possible dimensions.
(a) 2x – 3, 2x + 1
(b) 2x – 1, 2x + 3
(c) 3x + 1, 2x – 3
(d) 3x – 1, 2x + 3
Answer: b
Q47. For what value of b, is the polynomial x³ – 3x² + bx – 6 divisible by x – 3?
(a) 1
(b) 2
(c) 3
(d) -3
Answer: b
Q50. What is remainder when x³ – 2x² + x + 1 is divided by x – 1?
(a) 0
(b) -1
(c) 1
(d) 2
Answer: c
Q51. A polynomial with one degree is called:
a) Linear polynomial
b) Quadratic polynomial
c) Monomial
d) Binomial
Answer: a
Q52. √3 is a polynomial of degree:
a) 2
b) 0
c) 1
d) 1/2
Answer: b
Q53. Degree of the polynomial 7x5 + 8x2 – 5x + 3 is:
a) 1
b) 3
c) 2
d) 5
Answer: d
Q54. What is the degree of a zero polynomial?
a) 0
b) 1
c) Any natural number
d) Not defined
Answer: d
Q55. The value of the polynomial 7x4 + 3x2 – 4, when x = – 2 is:
a) 100
b) 110
c) 120
d) 130
Answer: c
Q56. The zero of the polynomial p(x) = -9x + 9is:
a) 0
b) -9
c) -1
d) 1
Answer: d
Q57. √12 X √15 is equal to:
a) 5√6
b) 6√5
c) 10√5
d) √25
Answer: b
Q58. If y97 + 97 is divided by y + 1, the remainder is:
a) 0
b) 1
c) 95
d) 96
Answer: d
Q59. If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is:
a) -3
b) 4
c) 2
d) -2
Answer: c
Q60. The value of 992 – 982 is:
a) 1
b) 197
c) 187
d) 207
Answer: b
Q61. One of the factors of (1 + 7x)2 + (49x2 – 1) is:
a) x – 7
b) 7 – x
c) 7x – 1
d) 14x
Answer: d
Q62. The factorization of 6x2 + 11x + 3 is:
a) (3x + 1) (2x + 3)
b) (x + 1) (2x + 3)
c) (x + 3) (2x + 1)
d) (3x + 3) (x + 1)
Answer: a
Q63. Which one of the following is a polynomial?
Answer: c
Q64. √3 is a polynomial of degree:
a) 2
b) 0
c) 1
d) 1/2
Answer: b
Q65. Degree of the polynomial 7x5 + 8x2 – 5x + 3 is:
a) 1
b) 3
c) 2
d) 5
Answer: d
Q66. What is the degree of a zero polynomial?
a) 0
b) 1
c) Any natural number
d) Not defined
Answer: d
Q67. The value of the polynomial 7x4 + 3x2 – 4, when x = – 2 is:
a) 100
b) 110
c) 120
d) 130
Answer: c
Q68. The zero of the polynomial p(x) = -9x + 9is:
a) 0
b) -9
c) -1
d) 1
Answer: d
Q69. √12 X √15 is equal to:
a) 5√6
b) 6√5
c) 10√5
d) √25
Answer: b
Q70. If y97 + 97 is divided by y + 1, the remainder is:
a) 0
b) 1
c) 95
d) 96
Answer: d
Q71. If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is:
a) -3
b) 4
c) 2
d) -2
Answer: c
Q72. The value of 992 – 982 is:
a) 1
b) 197
c) 187
d) 207
Answer: b
Q73. One of the factors of (1 + 7x)2 + (49x2 – 1) is:
a) x – 7
b) 7 – x
c) 7x – 1
d) 1+7X
Answer: d
Q74. The factorization of 6x2 + 11x + 3 is :
a) (3x + 1) (2x + 3)
b) (x + 1) (2x + 3)
c) (x + 3) (2x + 1)
d) (3x + 3) (x + 1)
Answer: a
Q75.
a) 1
b) –1
c) 0
d) 1/2
Answer: c
Q76.
a) 0
b) 1/4
c) 1/2
d) 1/√2
Answer: b
Q77. A binomial of degree 20 in the following is:
a. 20x + 1
b. x/20 + 1
c. x20 +1
d. x2+20
Answer: c
Explanation: A polynomial having two terms and the highest degree 20 is
called a binomial of degree 20.
Q78. x2 – x is ________ polynomial.
a. Linear
b. Quadratic
c. Cubic
d. None of the above
Answer: b
Explanation: A polynomial of degree two is known as quadratic polynomial.
Q79. 1+3x is a _________ polynomial.
a. Linear
b. Quadratic
c. Cubic
d. None of the above
Answer: a
Explanation: A polynomial of degree one is known as linear polynomial.
Q80. (x + 8)(x – 10) in the expanded form is:
(a) x2 – 8x – 80
(b) x2 – 2x – 80
(c) x2 + 2x + 80
(d) x2 – 2x + 80
Q81. The value of 95 x 96 is:
(a) 9020
(b) 9120
(c) 9320
(d) 9340
Q82.. The value of 104 x 96 is:
(a) 9984
(b) 9624
(c) 9980
(d) 9986
Q83. Without actual calculating the cubes the value of 283 + (–15)3 +(–13)3 is:
(a) 16380
(b) –16380
(c) 15380
(d) –15380
Q84. If x – 2 is a factor of x3 – 2ax2 +ax – 1 then the value of a is:
(a) 7/6
(b) –7/6
(c) 6/7
(d) 6/7
Q85. If x + 2 is a factor of x3 + 2ax2 +ax – 1 then the value of a is:
(a) 2/3
(b) 3/5
(c) 3/2
(d) 1/2
Q86. If x + y + z = 0 then x3 + y3 + z3 is equal to
(a) 3xyz
(b) – 3xyz
(c) xy
(d) –2xy
Q87. The factors of 2x2 – 7x + 3 are:
(a) (x – 3)(2x – 1)
(b) (x + 3)(2x + 1)
(c) (x – 3)(2x + 1)
(d) (x + 3)(2x – 1)
Q88. The factors of 6x2 + 5x – 6 are:
(a) (2x – 3)(3x – 2)
(b) (2x – 3)(3x + 2)
(c) (2x + 3)(3x – 2)
(d) (2x + 3)(3x + 2)
Q89. The factors of 3x2 – x – 4 are:
(a) (3x – 4)(x – 1)
(b) (3x – 4)(x + 1)
(c) (3x + 4)(x – 1)
(d) (3x + 4)(x + 1)
Q90. The factors of 12x2 – 7x + 1 are:
(a) (4x – 1)(3x – 1)
(b) (4x – 1)(3x + 1)
(c) (4x + 1)(3x – 1)
(d) (4x + 1)(3x + 1)
Q91. The factors of x3 – 2x2 – x + 2 are:
(a) (x – 1)(x – 1)(x – 5)
(b) (x + 1)(x + 1)(x + 5)
(c) (x + 1)(x – 1)(x + 5)
(d) (x + 1)(x + 1)(x – 5)
Q92. Factorise : 8a3+ b3 + 12a2b + 6ab2
(a) (2a + b) (2a + b) (2a + b)
(b) (2a – b) (2a – b) (2a + b)
(c) (2a-b)(2a-b)(2a-b)
(d) (-2a + b) (-2a + b) (2a + b)
Answer: a
Q93. The value of k for which x + 1 is a factor of the polynomial
x3 + x2 + x + k is :–
(a) 0
(b) 2
(c) 1
(d) -1
Answer: c
Q94. Using Remainder Theorem find the remainder when x3 – x2 + x – 1
is divided by x – 1
(a) -2
(b) -1
(c) 1
(d) 0
Answer: d
Q95. If a polynomial p(x) is divided by a linear divisor (x-a), then the remainder is
(a) p(a)
(b) p(1)
(c) p(0)
(d) p(x)
Answer: a
Q96. x-a is a factor of p(x) = ax2+bx+c. Which of the following is true?
(a) p(a) = 2
(b) p(a) = 0
(c) p(2) = 1
(d) p(b) = 0
Answer: b
Q97. Factorization of 1032-9 yields
(a) 10300
(b) 10600
(c) 11250
(d) 12500
Answer: b
98. The quadratic polynomial whose sum of zeroes is 3 and
the product of zeroes is –2 is :
(a) x2 + 3x – 2
(b) x2 – 2x + 3
(c) x2 – 3x + 2
(d) x2 – 3x – 2
Answer: d
Q99. Expansion of 2x(x + 2y) + 3x(2x – 3y) yields
(a) 8x2 – 5xy
(b) 8x – 5y
(c) 3y – 2y
(d) None of these
Answer: a
Q100. Factorize the quadratic polynomial by splitting the middle term:
y2 – 4 y –21
(a) (y – 7) (y – 3)
(b) (y – 7) (y + 3)
(c) (y + 7) (y – 3)
(d) (y + 7) (y + 3)
Answer: b
Q101. Factorise the quadratic polynomial by splitting the middle term:
x2 + 14x + 45
(a) (x + 9) (x+5)
(b) (x – 9) (x+5)
(c) (x-9) ( x-5)
(d) (x+9) ( x-5)
Answer: a
Q102. One of the linear factors of 3x2 + 8x + 5 is
(a) (x+1)
(b) (x-4)
(c) (x-2)
(d) (x+2)
Answer: a
Q103. The factors of 2x2 – 3x – 2 are :-
(a) (2x – 1) (x + 2)
(b) (2x + 1) (x – 2)
(c) (x + 1) (x – 2)
(d) (x – 1) (x + 2)
Answer: b
Q104. Factorize: (x – y)3 + (y – z)3 + (z – x)3
(a) (2x – y) (y – z) ( z – x)
(b) 3(x – y) (y – z) (z – x)
(c) (x – y) (2y – 3z) (z – x)
(d) (2x – 3y) (5x + 3y) (6x – 7y)
Answer: b
Q105. If 2x + 3y = 12 and xy = 5, find the value of 4x2 + 9y2.
(a) 67
(b) 98
(c) 56
(d) 84
Answer: d
Q106. Factorise: x2– 81
(a) (x+9)(x+9)
(b) (x-9)(x-9)
(c) (x-9)(x+9)
(d) (x-81)(x+1)
Answer: c
Q107. Find p(1/3) for p(t) = t2 – t + 2
(a) 22/9
(b) 15/9
(c) 16/9
(d) 14/9
Answer: c
Q108. Find the value of 5252 – 4752.
(a) 100
(b) 10000
(c) 50000
(d) 100000
Answer: c
Q109. Find the value of k if x2 + kx + 6 = (x + 2) (x + 3) for all k.
(a) 1
(b) -1
(c) 5
(d) 3
Answer: c
Q110. Find the value of p for which x + p is a factor of x2 + px + 3 – p.
(a) 1
(b) -1
(c) 3
(d) -3
Answer: c
Q111. If the area of the rectangle is 4x2 + 4x – 3, then find its possible dimensions.
(a) 2x – 3, 2x + 1
(b) 2x – 1, 2x + 3
(c) 3x + 1, 2x – 3
(d) 3x – 1, 2x + 3
Answer: b
MATCHING QUESTIONS
DIRECTION : In the section, each question has two matching lists. Choices for the correct combination of elements from Column-I and Column-II are given as options (a), (b), (c) and (d) out of which one is correct.
Column-II shows the degree of polynomials given Column-I
| Column-I | Column-II | ||
| ( P ) | 2 − 2 − 3 + 28 | (1) | 2 |
| ( Q ) | 2 | (2) | 1 |
| (R) | 5 − √7 | (3) | 0 |
| (S) | 4 − 2 | (4) | 8 |
| P | ||||
| (a) | 2 | 1 | 4 | 3 |
| (b) | 4 | 1 | 3 | 2 |
| (c) | 1 | 4 | 3 | 2 |
| (d) | 3 | 2 | 4 | 1 |
Ans : − 4, − 3, − 2, − 1
- The highest power of the variable is 8. So, the degree of the polynomial is 8 .
- The only term here is 2 which can be written as 20. So, the exponent of is 0 . Therefore, the degree of the polynomial is 0 .
- The highest power of the variable is 1. So, the degree of the polynomial is 1 .
- The highest power of the variable is 2. So, the degree of the polynomial is 2 . 6. Columa-II gives value of for polymomials given in
| Columa-I | Coluanu-II | ||
| () | 2 − 3 + | (1) | −2 |
| (Q) | 2 + + | (2) | 3/2 |
| (R) | 22 + + √2 | (3) | √2 − 1 |
| (S) |  |
(4) | −(2 + √2) |
: − 2, − 1, − 4, −
5. ASSERTION AND REASON DIRECTION : In each of the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
(a)Both assertion and reason are true and reason is the correct explanation of assertion.
(b)Both assertion and reason are true but reason is not the correct explanation of assertion.
(c) Assertion is true but reason is false.
(d) Assertion is false but reason is true.
1. Assertion : If () = 37 − 46 + + 9 is a polynomial, then its degree is 7 .
Reason : Degree of a polynomial is the highest power of the variable in it.
Ans : (a) Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
2. Assertion : ( + 2) and ( − 1) are factors of the polynomial 4 + 3 + 22 + 4 − 8
Reason : For a polynomial () of degree ≥ 1, − is a factor of the polynomial () if and only if () ≥ 1.
Ans : (c) Assertion is true but reason is false.
| () | = 4 + 3 + 22 + 4 − 8 |
| (−2)
So, ( + 2) is a factor of () |
= (−2)4 + (−2)3 + 2(−2)2 + 4(−2) − 8
= 16 − 8 + 8 − 8 − 8 = 0 |
(1) = (1)4 + (1)3 + 2(1)2 + 4(1) − 8
= 1 + 1 + 2 + 4 − 8 = 0
( − 1) is a factor of ()
3. Assertion : 32 + − 1 = ( + 1)(3 − 2) + 1.
Reason : If () and () are two polynomials such that degree of () ≥ degree of () and () ≥ 0 then we can find polynomials () and () such that
p() = ()() + (), where () = 0 of degree of () < degree of ()
Ane = (a) Both aeecrtion and reabon are true and reason is the correct explanation of assertion.
4. Assertion : The expression 34 − 43/2 + 2 = 2 is not a polynomial because the term −43/2 contains a rational power of .
Reason : The highest exponent in various terms of an algebraic expression in one variable is called its degree.
Ans: (b) Both assertion and reason are true but reason is not the correct explanation of assertion.
5. Assertion : ( + 2) is a factor of 3 + 32 + 5 + 6 and of 2 + 4
Reason : If () be a polynomial of degree greater than or equal to one, then ( − ) is a factor of () if () = 0
Ans: (a) Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
6. Assertion : The remainder when () = 3 − 62 + 2 − 4 is divided by 
Reason : If a polynomial () is divided by − , the remainder is the value of 
Ans: (a) Both assertion and reason are true and reason is the correct explanation of assertion.
8. Assertion : If ( + 1) is a factor of () = 2 + + 2, then = −3
Reason : If ( − ) is a factor of (), if () = 0.
Ans : (d) Assertion is incorrect but Reason is correct.
9. Assertion : If () = 4 + 3 − 22 + + 1 is divided by ( − 1), then its remainder is 2
Reason : If () be a polynomial of degree greater than or equal to one, divided by the linear polynomial − , then the remainder is (−)
Ans : (c) Assertion is correct but Reason is incorrect.
10. Assertion : The degree of the polynomial ( − 2)( − 3)( + 4) is 4
Reason : The mumber of zeroes of a polynomial is the degree of that polynomial.
Ans: (d) Assertion is false but reason is true.
() = ( − 2)( − 3)( + 4)
= ( − 2)[2 + 4 − 3 − 12]
= ( − 2)(2 + − 12)
= 3 + 2 − 12 − 22 − 2 + 24
() = 3 − 2 − 14 + 24
So, degree of () = 3
11. Assertion : If () = + , ≠ 0 is a linear polynomial, then 
is the only zero of () Reason : A linear polynomial has one and only one zero.
Ans : (a) Both Assertion and Reason are correct and Resson is the correct explanation of Assertion.