Polynomial MCQs With Answer

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Polynomial MCQs With Answer tailored for B. Pharm students help bridge algebraic foundations and pharmacy applications. This concise, exam-focused resource covers polynomial definitions, degrees, zeros, factor and remainder theorems, Vieta’s relations, synthetic division, and solving polynomial equations relevant to pharmaceutics and pharmacokinetics. Questions emphasize practical uses such as modeling drug-release kinetics, formulation calculations, and data fitting, while reinforcing problem-solving skills required in calculus and pharmaceutical mathematics. Each multiple-choice item includes clear options and explanations to boost accuracy, speed and conceptual depth. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the degree of the polynomial 3x^4 + 2x – 1?

  • 4
  • 3
  • 1
  • 0

Correct Answer: 4

Q2. Which of the following is a polynomial?

  • 1/x + 2
  • 3x^2 – 5x + 7
  • √x + 1
  • e^x

Correct Answer: 3x^2 – 5x + 7

Q3. The leading coefficient of 5x^3 – 2x + 9 is?

  • -2
  • 9
  • 5
  • 3

Correct Answer: 5

Q4. If f(x) is divisible by (x – 2), what is f(2)?

  • 2
  • -2
  • 0
  • 1

Correct Answer: 0

Q5. By the remainder theorem, the remainder of f(x) divided by (x – a) is?

  • f'(a)
  • a
  • f(a)
  • 0

Correct Answer: f(a)

Q6. The factor theorem states that (x – r) is a factor of f(x) if and only if:

  • f(r) = 0
  • f'(r) = 0
  • r = 0
  • Degree of f(x) = r

Correct Answer: f(r) = 0

Q7. Which statement about polynomial roots and degree is true?

  • Number of distinct real roots always equals degree
  • Total number of complex roots (counting multiplicity) equals degree
  • Polynomial degree is always greater than number of roots
  • Polynomials have infinite roots

Correct Answer: Total number of complex roots (counting multiplicity) equals degree

Q8. For the quadratic ax^2 + bx + c = 0, sum of roots is given by:

  • c/a
  • -b/a
  • b/a
  • -c/a

Correct Answer: -b/a

Q9. The product of the roots of ax^2 + bx + c = 0 is:

  • b/a
  • c/a
  • -c/a
  • -b/a

Correct Answer: c/a

Q10. If (x – 1)^2 is a factor of f(x), the root x = 1 has which multiplicity?

  • 0
  • 1
  • 2
  • 3

Correct Answer: 2

Q11. For large |x|, which term primarily determines the behavior of a polynomial?

  • Constant term
  • Lowest degree term
  • Leading term
  • Middle coefficient

Correct Answer: Leading term

Q12. If a polynomial has real coefficients, how do non-real complex roots occur?

  • They come alone
  • They appear in conjugate pairs
  • Only purely imaginary roots can occur
  • They cannot occur

Correct Answer: They appear in conjugate pairs

Q13. Which expression is NOT a polynomial?

  • 2x^3 + x – 5
  • x^(1/2) + 3
  • 4 – 6x
  • 0

Correct Answer: x^(1/2) + 3

Q14. Sum of coefficients of polynomial f(x) is equal to:

  • f(0)
  • f(1)
  • f(-1)
  • Derivative at 1

Correct Answer: f(1)

Q15. Alternating sum of coefficients a_n – a_{n-1} + … equals:

  • f(1)
  • f(-1)
  • f(0)
  • f'(1)

Correct Answer: f(-1)

Q16. Degree of product of two nonzero polynomials of degrees m and n is:

  • m + n
  • m – n
  • max(m, n)
  • min(m, n)

Correct Answer: m + n

Q17. If f(x) = x^3 – 6x^2 + 11x – 6, which is a root?

  • 1
  • 2
  • 3
  • All of these

Correct Answer: All of these

Q18. How many turning points can a degree 4 polynomial have at most?

  • 2
  • 3
  • 4
  • 5

Correct Answer: 3

Q19. Descartes’ Rule of Signs gives information about:

  • Number of negative roots exactly
  • Upper bound on positive real roots
  • Sum of complex roots
  • Degree of polynomial

Correct Answer: Upper bound on positive real roots

Q20. If f(x) has a repeated root r (multiplicity >1), then r is also a root of:

  • f'(x)
  • f”(x) only
  • Neither derivative
  • f(x) shifted by 1

Correct Answer: f'(x)

Q21. Which is the factorization of x^3 – a^3?

  • (x – a)(x^2 + ax + a^2)
  • (x + a)(x^2 – ax + a^2)
  • (x – a)^3
  • (x – a)(x^2 – ax + a^2)

Correct Answer: (x – a)(x^2 + ax + a^2)

Q22. The derivative of a polynomial of degree n has degree:

  • n + 1
  • n
  • n – 1
  • Depends on coefficients

Correct Answer: n – 1

Q23. Which method is fastest for dividing a polynomial by (x – c) in practical exams?

  • Long division only
  • Synthetic division
  • Grouping then factoring
  • Graphical method

Correct Answer: Synthetic division

Q24. If f(x) = x^2 – 5x + 6, factorized form is:

  • (x – 2)(x – 3)
  • (x + 2)(x + 3)
  • (x – 1)(x – 6)
  • (x + 1)(x + 6)

Correct Answer: (x – 2)(x – 3)

Q25. Sum of roots of cubic x^3 + px^2 + qx + r = 0 equals:

  • -p
  • p
  • -q
  • r

Correct Answer: -p

Q26. If f(x) is divided by (x^2 + 1), the remainder is:

  • A polynomial of degree at most 1
  • Always zero
  • A constant only
  • A polynomial of degree 2

Correct Answer: A polynomial of degree at most 1

Q27. Which of these polynomials is irreducible over the real numbers?

  • x^2 + 1
  • x^2 – 4
  • x^2 – x
  • x^2 + 2x + 1

Correct Answer: x^2 + 1

Q28. For f(x) = ax^2 + bx + c, f(1) gives:

  • a + b + c
  • a – b + c
  • a + b – c
  • a – b – c

Correct Answer: a + b + c

Q29. If polynomial f(x) has coefficients that sum to zero, what is true about x = 1?

  • f(1) = 0, so x = 1 is a root
  • x = 1 is not a root
  • f(-1) = 0
  • Constant term must be zero

Correct Answer: f(1) = 0, so x = 1 is a root

Q30. Which expression gives the remainder when f(x) is divided by (x + 2)?

  • f(-2)
  • f(2)
  • f'(2)
  • -f(2)

Correct Answer: f(-2)

Q31. Number of complex roots of x^4 + 1 = 0 is:

  • 0
  • 2
  • 4
  • 1

Correct Answer: 4

Q32. In pharmaceutical modeling, polynomial curve fitting is used primarily to:

  • Exactly represent non-polynomial data
  • Approximate experimental data trends
  • Replace mechanistic models always
  • Eliminate measurement error

Correct Answer: Approximate experimental data trends

Q33. If a polynomial p(x) satisfies p(0)=3 and p(1)=6, then sum of coefficients equals:

  • 3
  • 6
  • 9
  • Cannot be determined from given

Correct Answer: 6

Q34. Which property holds: f(-x) = -f(x) for a polynomial f means f is:

  • Even
  • Odd
  • Neither even nor odd
  • Constant

Correct Answer: Odd

Q35. If a polynomial has degree zero, it is:

  • Linear
  • Constant
  • Quadratic
  • Undefined

Correct Answer: Constant

Q36. Which operation can reduce the degree of a polynomial?

  • Multiplication by x
  • Differentiation
  • Addition of leading term
  • Multiplication by a nonzero constant

Correct Answer: Differentiation

Q37. For polynomial modeling of drug-release, increasing polynomial degree generally:

  • Always improves predictive power without drawback
  • Can overfit experimental noise
  • Makes model linear
  • Eliminates need for validation

Correct Answer: Can overfit experimental noise

Q38. Which of these is true about roots with odd multiplicity?

  • Graph touches x-axis and turns
  • Graph crosses the x-axis
  • Root is complex
  • Root cannot be 0

Correct Answer: Graph crosses the x-axis

Q39. Vieta’s formula relates polynomial coefficients to:

  • Values of derivative only
  • Sum and products of roots
  • Degree only
  • Leading coefficient only

Correct Answer: Sum and products of roots

Q40. If f(x) = (x – 2)(x – 3)(x – 5), constant term equals:

  • -30
  • 30
  • 0
  • 10

Correct Answer: -30

Q41. The quotient when x^2 + 3x + 2 is divided by x + 1 is:

  • x + 2
  • x + 1
  • x + 3
  • None of these

Correct Answer: x + 2

Q42. Which polynomial has x = i as a root (i = √-1) and real coefficients?

  • x + i
  • x^2 + 1
  • x^2 – 1
  • x – i

Correct Answer: x^2 + 1

Q43. If sum of roots of quadratic is 7 and product is 10, the quadratic (monic) is:

  • x^2 – 7x + 10
  • x^2 + 7x + 10
  • x^2 – 10x + 7
  • x^2 + 10x + 7

Correct Answer: x^2 – 7x + 10

Q44. Polynomial long division guarantees quotient and remainder with remainder degree less than:

  • Zero
  • Degree of divisor
  • Degree of dividend
  • Quotient degree

Correct Answer: Degree of divisor

Q45. Which expression represents a bivariate polynomial?

  • x^2 + 3
  • 2xy + y^2 + 1
  • √x + y
  • 1/x + y

Correct Answer: 2xy + y^2 + 1

Q46. If f(x) has integer coefficients and f(1/2) is rational non-integer, what does this imply about divisibility?

  • (2x – 1) is a factor
  • (x – 1/2) is not a factor with integer coefficients unless scaled
  • f has no rational roots
  • f must be constant

Correct Answer: (x – 1/2) is not a factor with integer coefficients unless scaled

Q47. Which identity is correct for sum of cubes?

  • a^3 + b^3 = (a + b)(a^2 – ab + b^2)
  • a^3 + b^3 = (a + b)^3
  • a^3 + b^3 = (a – b)(a^2 + ab + b^2)
  • a^3 + b^3 = (a + b)(a^2 + ab + b^2)

Correct Answer: a^3 + b^3 = (a + b)(a^2 – ab + b^2)

Q48. If sum of coefficients of polynomial p(x) equals 0, which factor must p(x) have?

  • (x + 1)
  • (x – 1)
  • (x)
  • (x^2 + 1)

Correct Answer: (x – 1)

Q49. Which is a common use of polynomials in pharmaceutics?

  • Modeling drug-release profiles by curve fitting
  • Measuring pH directly
  • Sterilization process control only
  • Counting tablet numbers physically

Correct Answer: Modeling drug-release profiles by curve fitting

Q50. If f(x) = x^4 – 5x^2 + 4, how many real roots does f(x)=0 have?

  • 0
  • 2
  • 4
  • 1

Correct Answer: 4

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