Please refer to Polynomials MCQ Questions Class 10 Mathematics below. These MCQ questions for Class 10 Mathematics with answers have been designed as per the latest NCERT, CBSE books, and syllabus issued for the current academic year. These objective questions for Polynomials will help you to prepare for the exams and get more marks.

**Polynomials MCQ Questions Class 10 Mathematics**

Please see solved MCQ Questions for Polynomials in Class 10 Mathematics. All questions and answers have been prepared by expert faculty of standard 10 based on the latest examination guidelines.

**MCQ Questions Class 10 Mathematics Polynomials**

**Question. If α,β are the zeros of the polynomial x ^{2} − px + q, then α^{2}/β^{2} +β^{2}/α^{2} is equal to − **

(a) p

^{4}/q

^{2}+2 −4p

^{2}/q

(b) p

^{4}/q

^{2}−2 +4p

^{2}/q

(c) p

^{4}/q

^{2}+2q

^{2 }−4p

^{2}/q

(d) None of these

**Answer**

A

**Question. If a,b are the zeros of the polynomial x ^{2} − px + 36 and α^{2} + β^{2} = 9, then p = **

(a) ± 6

(b) ± 3

(c) ± 8

(d) ± 9

**Answer**

D

**Question. If α,β are zeros of ax ^{2} + bx + c, ac ≠ 0, then zeros of cx^{2} + bx + a are − **

(a) − α − β

(b) α,1/ β

(c) β,1/ α

(d) 1/α,1 /β

**Answer**

D

**Question. If x − 3 is a factor of x ^{3} + 3x^{2} + 3x + p, then the value of p is **

(a) 0

(b) −63

(c) 10

(d) None

**Answer**

B

**Question. If 2 and 3 are the zeros of f(x) = 2x ^{3} + mx^{2} − 13x + n, then the values of m and n are respectively − **

(a) −5, − 30

(b) −5, 30

(c) 5, 30

(d) 5, − 30

**Answer**

B

**Question. If a,b are the zeros of the polynomial 6x ^{2} + 6px + p^{2}, then the polynomial whose zeros are (α + β)^{2} and (α −β)^{2} is − **

(a) 3x

^{2}+ 4p

^{2}x + p

^{4}

(b) 3x

^{2}+ 4p

^{2}x − p

^{4}

(c) 3x

^{2}− 4p

^{2}x + p

^{4}

(d) None of these

**Answer**

C

**Question. The value of ax ^{2} + bx + c when x = 0 is 6. The remainder when dividing by x + 1 is 6. The remainderwhen dividing by x + 2 is 8. Then the sum of a, b and c is **

(a) 0

(b) −1

(c) 10

(d) None

**Answer**

A

**Question. If the G.C.D. of the polynomials x ^{3} − 3x^{2} + px + 24 and x^{2} −7x + q is (x − 2), then the value of (p + q) is: **

(a) 0

(b) 20

(c) −20

(d) 40

**Answer**

A

**Question. The remainder of x ^{4} + x^{3 }− x^{2} + 2x + 3 when divided by x − 3 is **

(a) 105

(b) 108

(c) 10

(d) None

**Answer**

B

**Question. If the L.C.M. of two polynomials p(x) and q(x) is (x + 3)(x − 2) ^{2}(x − 6) and their H.C.F. is (x − 2).If p(x) = (x + 3)(x − 2)^{2}, then q(x) =________ **

(a) (x + 3)((x − 2)

(b) x2 − 3x − 18

(c) x2 − 8x + 12

(d) none of these

**Answer**

C

**Question. If α,β,γ are the zeros of the polynomial x ^{3} + 4x + 1, then (α + β)^{−1} + (β + γ)^{−1} + (γ + α)^{−1} = **

(a) 2

(b) 3

(c) 4

(d) 5

**Answer**

C

**Question. If α,β are the roots of ax ^{2} + bx + c and a + k, b + k are the roots of px^{2} + qx + r, then k = **

(a) −1/2[a/b−p/q]

(b) [a/b−p/q]

(c) 1/2[b/a−q/p]

(d) (ab − pq)

**Answer**

C

**Question. The condition that x ^{3} − ax^{2} + bx − c = 0 may have two of the roots equal to each other but of opposite signs is : **

(a) ab = c

(b) 2/3,a

(c) a

^{2}b = c

(d) None

**Answer**

A

**Question. The remainder when x1999 is divided by x ^{2} − 1 is **

(a) − x

(b) 3x

(c) x

(d) None

**Answer**

C

**Question. For the expression f(x) = x ^{3} + ax^{2} + bx + c, if f(a) = f(b) = 0 and f(d) = f(0). The values of a, b & c are **

(a) a = − 9, b = 20, c = − 12

(b) a = 9, b = 20, c = 12

(c) a = − 1, b = 2, c = − 3

(d) None of these

**Answer**

A

**Question. If x ^{2} − ax + b = 0 and x^{2} − px + q = 0 have a root in common and the second equation has equal roots, then **

(a) b + q = 2ap

(b) b + q = ap /2

(c) b + q = ap

(d) None of these

**Answer**

B

**Question. If f(x) = ax ^{2} + bx + c is divided by (bx + c), then the remainder is_____. **

(a) c

^{2}/b

^{2}

(b) ac

^{2}/b

^{2}+ 2c

(c) f(−c/b)

(d) ac

^{2}+2b

^{2}c/b

^{2}

**Answer**

C

**Question. ax ^{4} + bx^{3} + cx^{2} + dx + e is exactly divisible by x^{2} − 1, when: **

(a) a + b + c + e = 0

(b) a + c + e = 0

(c) a + b = 0

(d) a + c + e = b + d = 1

**Answer**

B

**Question. If x + 1 is a factor of ax ^{4} + bx^{3} + cx^{2} + dx + e = 0 then ____ **

(a) a + c + e = b + d

(b) a + b = c + d

(c) a + b + c + d + e = 0

(d) a + c + b = d + e

**Answer**

A

**Question. If a ^{3} −3a^{2}b + 3ab^{2} − b^{3} is divided by (a − b), then the remainder is **

(a) a

^{2}− ab + b

^{2}

(b) a

^{2}+ ab + b

^{2}

(c) 1

(d) 0

**Answer**

D

**Question. x ^{n} − y^{n} is divisible by x + y, when n is_______. **

(a) An odd positive integer

(b) An even positive integer

(c) An integer

(d) None of these

**Answer**

C

**Question. If α,β are the zeros of the quadratic polynomial 4x ^{2} − 4x + 1, then α^{3} + β^{3} is − **

(a) 1/ 4

(b) 1/ 8

(c) 16

(d) 32

**Answer**

A

**Question. If a + b = 4 and a ^{3} + b^{3} = 44, then a,b are the zeros of the polynomial. **

(a) 2x

^{2}− 7x + 6

(b) 3x

^{2}+ 9x + 11

(c) 9x

^{2 }− 27x + 20

(d) 3x

^{2}− 12x + 5

**Answer**

D

**Question. If the sum of zeros of the polynomial p(x) = kx ^{3 }− 5x^{2} − 11x − 3 is 2, then k is equal to **

(a) k = − 5/ 2

(b) k = 2/ 5

(c) k = 10

(d) k = 5/ 2

**Answer**

D

**Question. If y = f(x) = mx + c; then f(y) in terms of x is **

(a) mx + m + c

(b) m + mc + c

(c) m^{2}x + mc + c

(d) m^{2}x + m^{2}c

**Answer**

C

**Question. If 7 + 3x is a factor of 3x ^{3} + 7x, then the remainder is **

(a) 490/ 9

(b) −490/ 9

(c) 470/ 9

(d) None

**Answer**

B

**Question. The remainder when f(x) = 3x ^{4} + 2x^{3} −x2/3 −x/9 + 2/27 is divided by g(x) = x + 2/ 3 is **

(a) −1

(b) 1

(c) 0

(d) −2

**Answer**

C

**Question. f(x) = 3x ^{5} + 11×4 + 90x^{2 }− 19x + 53 is divided by x + 5 then the remainder is ______. **

(a) 100

(b) −100

(c) −102

(d) 102

**Answer**

C

**Question. If f(x) = 4x ^{3} − 6x^{2} + 5x − 1 and α,β and γ are its zeros, then αβγ = **

(a) 3/ 2

(b) 5/ 4

(c) −3/2

(d) 1/ 4

**Answer**

D

**Question. Consider f(x) = 8x ^{4} − 2x^{2} + 6x − 5 and a,b,g,d are it’s zeros then α + β + γ + δ = **

(a) 1/ 4

(b) − 1/ 4

(c) −3/2

(d) None

**Answer**

D

**Question. If (x − 3), (x −3) are factors of x ^{3 }− 4x^{2} − 3x + 18; then the other factor is **

(a) x + 2

(b) x + 3

(c) x − 2

(d) x + 6

**Answer**

A

**Question. The remainder when 1 + x + x ^{2} + x^{3} + ……….+ x2006 is divided by x− 1 is **

(a) 2005

(b) 2006

(c) 2007

(d) 2008

**Answer**

C

**Question. If (x − 1), (x + 1) and (x − 2) are factors of x ^{4} + (p − 3)x^{3} − (3p − 5)x^{2} + (2p − 9) x + 6 then the value of p is **

(a) 1

(b) 2

(c) 3

(d) 4

**Answer**

D

**Question. If f (−3/4)=0; then for f(x), which of the following is a factor? **

(a) 3x − 4

(b) 4x + 3

(c) −3x + 4

(d) 4x − 3

**Answer**

B

**Question. If the remainder when the polynomial f(x) is divided by x − 1, x + 1 are 6, 8 respectively then the remainder when f(x) is divided by (x − 1)(x + 1) is **

(a) 7 − x

(b) 7 + x

(c) 8 − x

(d) 8 + x

**Answer**

A

**Question. If a, b and g are the zeros of the polynomial f(x) = x ^{3} + px^{2} − pqrx + r, then 1/αβ + 1/βγ +1/γα = **

(a) r/p

(b) p/ r

(c) − p/ r

(d) − r/ p

**Answer**

B

**Question. The coefficient of x in x ^{2} + px + q was taken as 17 in place of 13 and it’s zeros were found to be − 2 and − 15. The zeros of the original polynomial are **

(a) 3, 7

(b) − 3, 7

(c) − 3, − 7

(d) −3, − 10

**Answer**

D

**Question. Let a, b be the zeros of the polynomial x ^{2} − px + r and ,α/2, 2β be the zeros of x^{2}− qx + r. Then the value of r is − **

(a) 2/9 (p−q)(2q−p)

(b) 2/9 (q−p)(2p−q)

(c) 2/9 (q −2p)(2q−p)

(d) 2/9(2p−q)(2q−p)

**Answer**

D

**Question. When x200 + 1 is divided by x ^{2} + 1, the remainder is equal to − **

(a) x + 2

(b) 2x − 1

(c) 2

(d) − 1

**Answer**

C

**Question. If a (p + q) ^{2} + 2bpq + c = 0 and also a(q + r)^{2} + 2bqr + c = 0 then pr is equal to − **

(a) p

^{2}+ a/ c

(b) q

^{2}+c/a

(c) p

^{2}+q/b

(d) q

^{2}+a/c

**Answer**

B

**Question. If α, β and γ are the zeros of the polynomial f(x) = ax ^{3} + bx^{2} + cx + d, then 1/α +1/β +1/γ = **

(a) −b/ a

(b) c/ d

(c) −c/d

(d) −c / a

**Answer**

C

**Question. If a, b and g are the zeros of the polynomial f(x) = ax ^{3 }− bx^{2} + cx − d, then α^{2} +β^{2} + γ^{2} = **

(a) b

^{2}−ac/a

^{2}

(b) b

^{2}+2ac/b

^{2}

(c) b

^{2}−2ac/a

(d) b

^{2}−2ac/a

^{2}

**Answer**

D

**Question. If a,b and c are not all equal and a and b be the zeros of the polynomial ax ^{2} + bx + c, then value of (1 + a + a^{2}) (1 + b + b^{2}) is : **

(a) 0

(b) positive

(c) negative

(d) non-negative

**Answer**

D

**Question. f(x) = 16x ^{2} + 51x + 35 then one of the factors of f(x) is **

(a) x − 1

(b) x + 3

(c) x − 3

(d) x + 1

**Answer**

D

**Question. Find the remainder obtained when x2007 is divisible by x ^{2} − 1. **

(a) x

^{2}

(b) x

(c) x + 1

(d) −x

**Answer**

B

**Question. If a polynomial 2x ^{3 }− 9x^{2} + 15x + p, when divided by (x − 2), leaves −p as remainder, then p is equal to **

(a) −16

(b) −5

(c) 20

(d) 10

**Answer**

B

**Question. If a, b and g are the zeros of the polynomial 2x ^{3} − 6x^{2} − 4x + 30, then the value of (ab + bg + ga) is **

(a) − 2

(b) 2

(c) 5

(d) − 30

**Answer**

A

**Question. If ax ^{3} + 9x^{2} + 4x − 1 is divided by (x + 2), the remainder is −6; then the value of ‘a’ is **

(a) −3

(b) −2

(c) 0

(d) 33 8

**Answer**

D

**Question. If c, d are zeros of x ^{2} − 10ax − 11b and a, b are zeros of x^{2} − 10cx − 11d, then value of a + b + c + d is **

(a) 1210

(b) − 1

(c) 2530

(d) − 11

**Answer**

A

**Question. If the ratio of the roots of polynomial x ^{2} + bx + c is the same as that of the ratio of the roots of x^{2} + qx + r, then **

(a) br

^{2}= qc

^{2}

(b) cq

^{2}= rb

^{2}

(c) q

^{2}c

^{2}= b

^{2}r

^{2}

(d) bq = rc

**Answer**

B

**Question. The quadratic polynomial whose zeros are twice the zeros of 2x ^{2} − 5x + 2 = 0 is − **

(a) 8x

^{2}− 10x + 2

(b) x

^{2}− 5x + 4

(c) 2x

^{2}− 5x + 2

(d) x

^{2}− 10x + 6

**Answer**

B

**Question. If α,β,γ are the zeros of the polynomial x ^{3} − 3x + 11, then the polynomial whose zeros are (α+β), (β+γ) and (γ+α) is − **

(a) x

^{3}+ 3x + 11

(b) x

^{3}− 3x + 11

(c) x

^{3}+ 3x − 11

(d) x

^{3}− 3x − 11

**Answer**

D

**Question. If α,β,γ are such that α + β + γ = 2, α ^{2} + β^{2} + γ^{2} = 6, α^{3} + β^{3} + γ^{3} = 8, then α4 + β^{4} + γ^{4} is equal to **

(a) 10

(b) 12

(c) 18

(d) None

**Answer**

C

**Question. If one zero of the polynomial ax ^{2} + bx + c is posi- tive and the other negative then (a,b,c ∈R, a ≠ 0) **

(a) a and b are of opposite signs.

(b) a and c are of opposite signs.

(c) b and c are of opposite signs.

(d) a,b,c are all of the same sign.

**Answer**

B

**Question. A real number is said to be algebraic if it satisfies a polynomial equation with integral coefficients. Which of the following numbers is not algebraic : **

(a) 2/ 3

(b) 2

(c) 0

(d) π

**Answer**

D

**Question. If two zeroes of p(x)= 2x ^{4} + x^{3} – 14x^{2} – 19x – 6 are –1 and –2, the other two zeroes are: **

(a) 1/ 2 and – 3

(b) −1/ 2 and – 3

(c) −1/ 2 and 3

(d) none of these

**Answer**

C

**Question. The graph of y = – 3x ^{2} + 2x– 1 cuts the x-axis at : **

(a) 1/ 3 and 0

(b) 1/ 3 and 1/ 3

(c) do not cut

(d) none of these

**Answer**

C

**Question. The two zeroes of f(x) = x ^{4}– 6x^{3} – 26x^{2} + 138x – 35 are 2 ± √3 , the other two zeroes are : **

(a) 7 and –5

(b) –7 and 5

(c) –7 and – 5

(d) none of these

**Answer**

A

**Question. The zeroes of y = x ^{2} + 7x + 12 are : **

(a) 4 and – 3

(b) –4 and –3

(c) –4 and 3

(d) 4 and 3

**Answer**

B

**Question. If the zeroes of f(x) = x ^{3} – 3x^{2} + x + 1 are a – b, a, a + b, then value of a and b is : **

(a) a =1, b = −√ 2

(b) a =1, b = −√2

(c) a =1, b =√2

(d) none of these

**Answer**

B

**Question. The zeroes of f (x) = 4 √3x ^{2} +5x − 2 √3 are : **

(a) 2/√3 and −√3/4

(b) −/√32 and −√3/4

(c) −2/√3 and √3/4

(d) none of these

**Answer**

D

**Question. The sum and product of zeroes of quadratic polynomial 3x ^{2} – 8x + 12 is : **

(a) −8/3 , 4

(b) 8/3 , 4

(c) −8/3 , −4

(d) none of these

**Answer**

B

**Question. The graph of y = f (x) is given in figure (ii). How many zeroes are there of f (x)? **

(a) 0

(b) 1

(c) 2

(d) many

**Answer**

B

**Question. The number of polynomials having zeroes as –2 and 5 is **

(a) 1

(b) 2

(c) 10

(d) infinite

**Answer**

D

**Question. The solution of x ^{2} + 6x + 9 = 0 is **

(a) –1

(b) 3

(c) –3

(d) 1

**Answer**

C

**Question. The graph of y = f (x) is given in figure (i), for some polynomial f (x). The number of zeroes of f (x) is **

(a) 1

(b) 2

(c) 3

(d) many

**Answer**

C

**Question. The value of k for which (–4) is a zero of the polynomial x ^{2} – x – (2k + 2) is **

(a) 2

(b) –6

(c) 9

(d) 8

**Answer**

C

**Question. The graph of y = f (x) is given in figure (iii). How many zeroesare there of f (x)? **

(a) 1

(b) 2

(c) 3

(d) None

**Answer**

D

**Question. The zeroes of the quadratic equation 4s ^{2} – 4s + 1 are **

(a) 1/2 , 1/4

(b) 1/2 , 1/2

(c) 1/4 , 1/14

(d) 1/3 , 1/4

**Answer**

B

**Question. The quadratic polynomial, sum of whose zeroes is 8 and their product is 12, is given by **

(a) x^{2} – 8x + 12

(b) x^{2} + 8x – 12

(c) x^{2} – 5x + 7

(d) x^{2} + 5x – 7

**Answer**

A

**Question. A quadratic polynomial whose product and sum of zeroes are −13/ 5 and 3/5 , respectively. **

(a) k(x^{2} + 12x + 5)

(b) k[x^{2} – (8x) + (–9))

(c) k [x^{2} − (1/2 x) + ( −7/5)]

(d) k [x^{2} − (3/5 x) + (−13/5)]

**Answer**

D

**Question. A quadratic polynomial whose zeroes are 1 and –3 is **

(a) x^{2} + 3x – 2

(b) x^{2} + 5x – 5

(c) x^{2} + 2x – 3

(d) None of these

**Answer**

C

**Question. The sum and product of zeroes of the quadratic equation given in example 3 are respectively **

(a) 2, 3/4

(b) 0, 1/8

(c) 1, 1/4

(d) None of these

**Answer**

C

**Question. The set of the zeroes of the polynomial x ^{2} – 25, their sum and product is **

(a) 4, 3; 7; 12

(b) –3, 3; 0; –9

(c) 5, –5; 0; –25

(d) None of these

**Answer**

C

**Question. If the product of the zeroes of the polynomial ax ^{2} – 6x – 6 is 4, then value of a is **

(a) 1/8

(b) – 1/4

(c) 5/3

(d) –3/2

**Answer**

D

**Question. If a, b are the zeroes of the polynomial 2x ^{2} – 5x + 7, then a polynomial whose zeroes are 2α + 3β, 3α + 2β is **

(a) k(x

^{2}− 3/5 x + 21)

(b) k (x

^{2}− 25/2 x + 41)

(c) k (x

^{2}+9/2 x − 45 )

(d) None of these

**Answer**

B

**Question. The value of k, if the sum of the zeroes of the polynomial x ^{2} – (k + 6) x + 2 (2k – 1) is half of their product is **

(a) 7

(b) 11

(c) 12

(d) None of these

**Answer**

A

**Question. A quadratic equation x ^{2} – 2x – 8 is given. The zeroes of it are **

(a) –2 and 4

(b) 3 and 5

(c) 1 and 6

(d) None of these

**Answer**

A

**Question. If the zeroes of the polynomial x ^{2} + px + q are double in value to the zeroes of 2x^{2} – 5x – 3, the values of p and q are respectively **

(a) 5, 6

(b) 4, 7

(c) –5, –6

(d) –4, –7

**Answer**

C

**Question. The sum and product of the zeroes of the quadratic equation given in example 1 are respectively **

(a) 2, 4

(b) 5, –8

(c) 6, 8

(d) 2, –8

**Answer**

D

**Question. The remainder when f (x) = – 3x ^{2} + x^{4} + 4x + 5 is divided by –x^{2} + 2 is **

(a) 6

(b) 8

(c) 12

(d) none of these

**Answer**

D

**Very Short Answer and Question :**

**Question. Find the common zero of x ^{2} –1, x^{4} –1 and (x –1)^{2} ? **

**Answer**

**x + 1**

**Question. Write a polynomial whose sum and product of zeroes are 2 and –9 respectively. **

**Answer**

**x ^{2} – 2x – 9**

**Question. If α, β are the zeroes of quadratic polynomial x ^{2} – 3x + 2, form a quadratic polynomial whose zeroes are –α, and –β . **

**Answer**

**k (x ^{2} + 3x + 2)**

**Question. Find the zeroes of the polynomial 9x ^{2} – 25. **

**Answer**

**± 5/3**

**Question. For what value of x, both the polynomials p(x) – x ^{2} – x – 6 and q(x) – x^{2} – 2x –15 becomes zero. **

**Answer**

** x = 3**

**Question. How many maximum zeroes will the polynomial 3x ^{3} + 6x^{2} – 7 can have ? **

**Answer**

**3**

**Question. What should be added to the polynomial p(x) = x ^{2} – 5x + 4, so that 2 is a zero of p(x)? **

**Answer**

** 2**

**Question. If α and β are the zeroes of the polynomial x2 – 5x + 6, find the value of 1/α +1/β . **

**Answer**

** 5/ 6**

**(a) If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.****(b) If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.****(c) If Assertion is correct but Reason is incorrect.****(d) If Assertion is incorrect but Reason is correct.**

**Question. Assertion : x ^{3} + x has only one real zero. **

**Reason : A polynomial of nth degree must have n real zeroes.**

**Answer**

C

**Question. Assertion : Degree of a zero polynomial is not defined. ****Reason: Degree of a non-zero constant polynomial is ‘0’.**

**Answer**

B

**Question. Assertion : Zeroes of f(x) = x ^{2} – 4x – 5 are 5, – 1. **

**Reason : The polynomial whose zeroes are 2 + √3, 2 –√3 is x**

^{2}– 4x + 7.**Answer**

C

**Question. Assertion : x ^{2} + 4x – 5 has two zeroes. **

**Reason : A quadratic polynomial can have at the most two zeroes.**

**Answer**

D

**Question. Assertion : If one zero of polynomial p(x) = (k ^{2} + 4) x^{2} + 13x + 4k is reciprocal of other, then k = 2. **

**Reason : If (x – α) is a factor of p(x), then p(α) = 0 i.e. α is a zero of p(x).**

**Answer**

B