Polynomials Questions and Answers

Problem questions on polynomials (Algebra):

Q 1. If (2x – 1) is a factor of both 6x² + ax – 4 and bx² – 11x + 3, then the values of a and b are
A. a = 5, b = 10
B. a = 3, b = 10
C. b = 2a, a = 4
D. a = 2b, b = 2

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Answer:-A. a = 5, b = 10

Q 2. The remainder has no term in x², when x⁵ + kx² is divided by (x – 1) (x – 2) (x – 3), then the value of  k is
A. 90
B. 80
C. –80
D. –90

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Answer:-D. –90

Q 3. If x + 1 is a factor of xⁿ + 1 then the value of n is
A. even
B. odd
C. any integer
D. none

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Answer:-B. odd

Q 4. The quadratic function in x, which when divided by (x – (a), (x – (b) and (x – (c) leaves remainders 1, 2 and 4 respectively, is
A. 1/2  x² – 1/2x + 1
B. x² – 1/2x + 2
C. 2x² + 1/2x + 1
D. None of these

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Answer:-A. 1/2  x2 – 1/2x + 1

Q 5. If 3x⁵ + 11x⁴ + 90x² – 19x + 53 is divided by x + 5 then the remainder is
A. 102
B. 104
C. –102
D. –104

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Answer:-C. –102

Q 6. If x² – 3x + 2 is one of the zero of x⁴ – px² + q then the values of p and q are
A. 4 and 6
B. 3 and 5
C. 5 and 6
D. 5 and 4

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Answer:-D. 5 and 4

Q 7. The remainder when x⁴ – y⁴ is divided by (x – y) is
A. 0
B. x + y
C. x² – y²
D. 2y²

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Answer:-A. 0

Q 8. The remainder, when x³ – 7x²a + 8xa² + 15a³ is divided by (x + 2(a) is
A.–35a³
B. –37a³
C. 37a³
D. 35 a³

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Answer:-B. –37a3

Q 9. The remainder, when 1 – x – xⁿ + x^{n + 1},  is divided by 1 – 2x + x² is
A. 1
B. 2
C. 5
D. zero

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Answer:-D. zero

Q 10. If ax² + bxy + cy² is a homogeneous expression of 2nd degree where 4a = b = c and a is a square of 2 then the expression is
A. 16x² + 16xy + 4y²
B. 16x² + 4xy + 16y²
C. 4x² + 16xy + 16y²
D. 16x² + 16xy + 16y²

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Answer:-C. 4×2 + 16xy + 16y2

Q 11. The multiple of 4x³ + 4x² – x – 1 in the following is
A. 2x + 1
B. 2x + 3
C. x – 1
D. x – 2

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Answer:-A. 2x + 1

Q 12. If f(x) = x⁴ – 2x³ + 3x² – ax + b is a polynomial such that when it is divided by x – 1 and x + 1, the remainders are respectively 5 and 19. Then the remainder when f(x) is divided by (x – 2) is
A. 5
B. 10
C. 15
D. 25

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Answer:-B. 10

Q 13. Without actual substitution, the value of  x⁶ – 19x⁵ + 69x⁴ – 151x³ + 229x² + 166x + 26, when x = 15 is
A. 15
B. 30
C. 40
D. 41

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Answer:-D. 41

Q 14. Without actual substitution, the value of 32x⁵ – 48x⁴ + 40x³ – 60x² + 26x – 38, when x  = 1.5 is
A. 0
B. 2
C. 1.5
D. 1

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Answer:-D. 1

Q 15. If (x + (a) is a factor of x³ + ax² – 2x + a + 4. Then the value of a is
A. –1/3
B. –2/3
C. 1
D. –4/3

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Answer:-D. –4/3

Q 16. Without actual division, x⁴ + 2x³ –2x² + 2x – 3 is exactly divisible by
A. x² – 2x + 3
B. x² – 2x – 3
C. x² – 2x – 3
D. x² + 2x – 3

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Answer:-D. x2 + 2x – 3

Q 17. What must be subtracted from 4x⁴ – 2x³ – 6x² + x – 5, so that the result is exactly divisible by 2x² + x – 1 is
A. – 3
B. – 5
C. – 6
D. – 8

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Answer:-C. – 6

Q 18. The degree of the polynomial (2x⁵ +  3x⁴ + 2x² – 10x + 1)3 is
A. 23
B. 5
C. 10
D. 15

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Answer:-D. 15

Q 19. Given that x² – 4x + 1 is a factor of x4 – 9x³ + 27x² – 29x + 6, then the other factors are
A. (x + 2), (x – 3)
B. (x – 2), (x + 3)
C. (x – 2), (x – 3)
D. (x + 2), (x + 3)

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Answer:-C. (x – 2), (x – 3)

Q 20. Sum of the values of the polynomial p(x) = x⁴ – 3x² + 2x – 1 at x = 1 and x = 2 is,
A. 4
B. 5
C. 6
D. 7

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Answer:-C. 6

Q 21. If a monomial, a binomial and a trinomial are added, then the minimum and maximum number of terms in the obtained polynomial are respectively
A. 1, 3
B. 1, 6
C. 3, 6
D. 3, 3

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Answer:-B. 1, 6

Q 22. The number that must be added to x³ – 3x² – 12x  + 19 so that 2 is a zero of the polynomial, is
A. 6
B. 8
C. 9
D. 12

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Answer:-C. 9

Q 23. Which of the following numbers is a multiple of zero of the polynomial x³ – x² – x – 2 ?
A. 125
B. 27
C. 16
D. None of these

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Answer:-C. 16

Q 24. If ‘–a’ is a zero of the polynomial xⁿ + aⁿ, then n is
A. even
B. odd
C. A or B
D. can’t say

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Answer:-B. odd

Q 25. Given that x² – 4x + 5 is a factor of x⁴ – 6x³ + 18x² – 30x + 25, then the remaining factor is
A. x² – 2x + 5
B. x² + 2x – 5
C. x² + 2x + 5
D. None of these

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Answer:-A. x2 – 2x + 5

Q 26. If A, B, C are the remainders of x³ – 3x² – x + 5, 3x⁴ – x³ + 2x²  – 2x – 4, 2x⁵ – 3x⁴ + 5x³ – 7x² + 3x – 4, when divided by (x + 1), (x + 2), (x – 2) respectively, then the ascending order of A, B, C is
A. A, C ,B
B. B, C, A
C. A, B ,C
D. B, A, C

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Answer:-A. A, C ,B

Q 27. If the quotient of x⁴– 11x³ + 44x² – 76x + 48, when divided by x² – 7x + 12 is Ax² + Bx + C, then the descending order of A, B, C is
A. A, B, C
B. B, C, A
C. A, C, B
D. C, A, B

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Answer:-D. C, A, B

Q 28. If (x + 1), (x – 1), (x – 3) are the factors of x³ + Ax² + Bx + C, then the ascending order of A, B, C is
A. A, B, C
B. B, C, A
C. A, C, B
D. B, A, C

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Answer:-A. A, B, C

Q 29. If ax + by is a homogenous expression of first degree in x and y. When x = 1, y = 2 the value of expression is 9 and when x = 2, y = 3 the value of equation is 16. Then the expression is
A. 2x + 5y
B. 5x + 2y
C. 5x + 4y
D. 5x + 3y

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Answer:-B. 5x + 2y

Q 30. The expression ax² + bx + c equals –2 when x = 0, leaves remainder 3 when divided by (x – 1) and a remainder –3 when divided by (x + 1). Then the values of a, b, c are respectively
A. 2, 3
B. –2, 3
C. 2, –3
D. –2, –3

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Answer:-A. 2, 3

Q 31. The quadratic polynomial in x, which when divided by (x – 1), (x – 2) and (x – 3) leaves remainders of 11, 22 and 37 is
A. 2x² – 5x + 4
B. 2x² + 5x + 4
C.  2x² + 5x – 4
D.  None of these

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Answer:-B. 2×2 + 5x + 4

Q 32. Given that x2 – 6x + 7 is a factor of x³ – 11x² + 37x – 35, then the other factor is
A. (x – 2)
B. (x – 3)
C. (x – 1)
D. (x – 5)

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Answer:-D. (x – 5)

Q 33. If ‘–2’ is a zero of the polynomial ax³ + bx² + x – b and the value of the polynomial is 4 at x = 2, then the values of a and b respectively are
A. – ¼, 0
B.  – ½, 0
C. – ½, ½
D. – ¼, ½

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Answer:-A. – ¼, 0

Q 34. If p = x² + xy + 2y², Q = x³ + xy² + y³, R = a³ + b³ + c³, S = a⁴ + 3a³b + 4a²b² are homogeneous expressions then the complete homogenous expression is
A. S
B. P
C. Q
D. R

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Answer:-D. R

Q 35. If A = x³ + 4x²y + 5xy² + y³, B = x³ + y³ + x² + y² + x + y, C = a²(b + c) + b²(c + (a) + c²(a² + b³), D = 2x² + 3y³ + x³ + y² + 1 are algebraic expressions, then symmetric expression is
A. A
B. B
C.D
D. E

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Answer:-B. B

Q 36. 2(x² + y²) + 3(x + y) + 7 is
A. homogeneous
B. symmetric
C. not symmetric
D. homogeneous symmetric

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Answer:-B. symmetric

Q 37. If 2x³ + 3x² + ax + b is divided by ( x – 2) leaves remainder 2 and divided by (x + 2) leaves remainder –2, then the values of a, b are respectively
A. 7, 12
B. –7, –12
C. 7, –12
D. –7,12

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Answer:-B. –7, –12

Q 38. What must be added to 3x³ + x² – 22x + 9 so that the result is exactly divisible by 3x² + 7x – 6 is
A. 2x – 3
B. 2x – 1
C. 2x + 1
D.  2x + 3

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Answer:-D.  2x + 3

Q 39. What must be subtracted from x³ – 6×2 – 15x + 80, so that the result is exactly divisible by x² + x – 12 is
A. 4(x – 1)
B. 4(x + 1)
C. 4(x + 2)
D. 4(x – 2)

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Answer:-A. 4(x – 1)

Q 40. If x² – 4 is a factor of ax⁴ + 2x³ – 3×2 + bx – 4, then the values of a and b are
A. a = 1, b = – 8
B. a = 3, b = 5
C. a = 6, b = 7
D. a = – 1, b = 5

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Answer:-A. a = 1, b = – 8

Q 41. If (x + (a) is a factor of x⁴ – a²x² + 3x – a, then the value of a is
A.1
B. 2
C. 3
D. 0

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Answer:-D. 0