Algebra becomes more interesting when students start understanding how mathematical expressions actually behave, and that is where Polynomials Class 10 MCQs play an important role. This chapter is one of the core algebra chapters in Class 10 Mathematics because it introduces students to polynomial expressions, zeroes of polynomials, graphical interpretation, and relationships between coefficients and roots. Many questions in school exams and CBSE Board papers are directly based on concept clarity rather than memorization, so practicing objective questions regularly becomes very useful. Students preparing for board exams should especially focus on quadratic polynomials, polynomial graphs, and division algorithm concepts because these areas are commonly included in competency-based patterns. Solving MCQs on Polynomials Class 10 also helps students improve analytical thinking, calculation accuracy, and algebraic understanding in a much better way.
Main Concepts Covered in Polynomials
Before attempting the questions, students should revise the important concepts from the chapter carefully. Most Class 10 Polynomials MCQs are framed from these core topics.
- Meaning of polynomial
- Types of polynomials
- Linear polynomial
- Quadratic polynomial
- Cubic polynomial
- Degree of polynomial
- Zeroes of polynomial
- Geometrical meaning of zeroes
- Relationship between zeroes and coefficients
- Graphical representation of polynomials
- Polynomial division algorithm
- Factorization concepts
A clear understanding of these topics helps students solve both direct and application-based questions more confidently.
Polynomials Class 10 MCQs with Answers
Solve important and exam-oriented Polynomials Class 10 MCQs designed according to the latest CBSE pattern and competency-based approach. These objective questions will help students strengthen concepts like zeroes of polynomials, polynomial graphs, coefficient relationships, and algebraic expressions while improving speed and accuracy for board exams.
Q. Which of the following is NOT a characteristic of a polynomial?
A) Involves variables and constants
B) Connected by addition, subtraction and multiplication operations
C) Powers of variables are always whole numbers
D) Powers of variables can be negative integers or fractions
Answer: D
Explanation: In a polynomial, powers of variables must be non-negative whole numbers only.
Q. A polynomial of degree one is classified as a:
A) Quadratic polynomial
B) Cubic polynomial
C) Linear polynomial
D) Biquadratic polynomial
Answer: C
Explanation: A polynomial with highest power one is called a linear polynomial.
Q. What does a “zero” of a polynomial signify?
A) The highest power of the variable in the polynomial
B) The constant term of the polynomial
C) The value of the variable for which the polynomial evaluates to zero
D) The coefficient of the highest power of the variable
Answer: C
Explanation: A zero of a polynomial is the value of the variable that makes the polynomial equal to zero.
Q. For a quadratic polynomial ax² + bx + c where a is not equal to zero, what is the relationship between the sum of its zeros and its coefficients?
A) α + β = c/a
B) α + β = -b/a
C) α + β = b/a
D) α + β = -c/b
Answer: B
Explanation: For quadratic polynomial ax² + bx + c, sum of zeros is equal to minus b divided by a.
Q. For a quadratic polynomial ax² + bx + c where a is not equal to zero, what is the relationship between the product of its zeros and its coefficients?
A) αβ = -b/a
B) αβ = a/c
C) αβ = c/a
D) αβ = -c/a
Answer: C
Explanation: Product of zeros of a quadratic polynomial is equal to c divided by a.
Q. How can the zeros of a polynomial be geometrically interpreted?
A) The points where the graph intersects the y-axis
B) The maximum or minimum points of the graph
C) The points where the graph intersects the x-axis
D) The slope of the graph at any point
Answer: C
Explanation: Zeros of a polynomial are the x-coordinates where the graph cuts or touches the x-axis.
Q. According to the Division Algorithm for polynomials, if P(x) is the dividend and G(x) is the divisor where G(x) is not zero, then P(x) can be expressed as:
A) P(x) = G(x) - R(x)
B) P(x) = G(x) + Q(x) + R(x)
C) P(x) = G(x)Q(x) + R(x)
D) P(x) = Q(x) / G(x) + R(x)
Answer: C
Explanation: Polynomial division algorithm states P(x) = G(x)Q(x) + R(x).
Q. Understanding polynomials is crucial for which of the following higher mathematical concepts?
A) Coordinate geometry
B) Calculus
C) Algebra
D) All of the above
Answer: D
Explanation: Polynomials are widely used in algebra, coordinate geometry, and calculus.
Q. Which of the following skills are enhanced by understanding polynomials?
A) Logical reasoning and graph interpretation
B) Equation solving ability
C) Graph interpretation and analytical thinking
D) All of the above
Answer: D
Explanation: Studying polynomials improves multiple mathematical and analytical skills.
Q. Identify the type of polynomial based on its degree: P(x) = 5x² + 2x - 7
A) Linear polynomial
B) Quadratic polynomial
C) Cubic polynomial
D) Constant polynomial
Answer: B
Explanation: Highest power of x is two, so it is a quadratic polynomial.
Q. If the graph of a polynomial intersects the x-axis at exactly two distinct points, how many real zeros does the polynomial have?
A) One
B) Two
C) Three
D) Cannot be determined
Answer: B
Explanation: Each x-axis intersection represents one real zero.
Q. A quadratic polynomial has zeros two and five. Which of the following could be the polynomial?
A) x² - 7x + 10
B) x² - 3x - 10
C) x² + 7x + 10
D) x² + 3x - 10
Answer: A
Explanation: Sum of zeros is seven and product is ten, giving polynomial x² - 7x + 10.
Q. When a polynomial P(x) is divided by (x - a), the remainder is P(a). This is a statement of:
A) Factor Theorem
B) Remainder Theorem
C) Division Algorithm
D) Fundamental Theorem of Algebra
Answer: B
Explanation: Remainder theorem states that remainder on division by (x - a) is P(a).
Q. Which of the following expressions is a polynomial?
A) x² + √x - 3
B) 1/x + 5
C) x³ - 4x + 7
D) x⁻² + 2x
Answer: C
Explanation: A polynomial cannot have fractional or negative powers of variables.
Q. If α and β are the zeros of the quadratic polynomial x² - 5x + 6, then find the value of α + β.
A) -5
B) 5
C) 6
D) -6
Answer: B
Explanation: Sum of zeros = minus coefficient of x divided by coefficient of x² = 5.
Q. If α and β are the zeros of the quadratic polynomial x² - 5x + 6, then find the value of αβ.
A) -5
B) 5
C) 6
D) -6
Answer: C
Explanation: Product of zeros = constant term divided by coefficient of x² = 6.
Q. A quadratic polynomial has a graph opening downwards if:
A) It has no real zeros
B) It has exactly one real zero
C) It has exactly two distinct real zeros
D) Its leading coefficient is negative
Answer: D
Explanation: If the coefficient of x² is negative, the parabola opens downward.
Q. If P(x) = x³ - 3x² + 5x - 3 is divided by G(x) = x² - 2, what is the degree of the quotient Q(x)?
A) 2
B) 1
C) 0
D) 3
Answer: B
Explanation: Degree of quotient = degree of dividend minus degree of divisor = 3 - 2 = 1.
Q. If the sum of the zeros of a quadratic polynomial kx² + 2x + 3k is equal to their product, find the value of k.
A) k = -2/3
B) k = 2/3
C) k = -1/3
D) k = 1/3
Answer: A
Explanation: Sum of zeros = -2/k and product = 3. Equating gives k = -2/3.
Q. Consider the graph of a polynomial P(x). If the graph touches the x-axis at a point but does not cross it, what does this indicate about the zero at that point?
A) It is a simple zero
B) It is a zero with an even multiplicity
C) It is a zero with an odd multiplicity
D) It is not a real zero
Answer: B
Explanation: A graph touching but not crossing the x-axis indicates even multiplicity.
Q. When P(x) = x² - 5x + 6 is divided by G(x) = x - 2, what is the remainder?
A) 5x + 10
B) -5x + 2
C) 5x - 10
D) 5x - 2
Answer: C
Explanation: The quotient after division is x - 3 and remainder is zero. The intended correct answer in the paper appears inconsistent.
Q. If one zero of the quadratic polynomial P(x) = ax² + bx + c is the reciprocal of the other, then which of the following is true?
A) a = b
B) a = -c
C) b = c
D) a = c
Answer: D
Explanation: Product of reciprocal zeros is one, so c/a = 1, therefore a = c.
Q. If (x + 1) is a factor of the polynomial 2x² + kx, then the value of k is:
A) -2
B) 2
C) 3
D) 5
Answer: B
Explanation: Using factor theorem, substituting x = -1 gives 2 - k = 0, so k = 2.
Q. If α and β are the zeros of the polynomial f(x) = x² + x + 1, then 1/α + 1/β is equal to:
A) 1
B) -1
C) 0
D) 2
Answer: B
Explanation: (α + β)/αβ = -1/1 = -1.
Q. What is the condition for the remainder R(x) in the Division Algorithm P(x) = G(x)Q(x) + R(x)?
A) Degree of R(x) greater than degree of G(x)
B) R(x) = 0 only
C) Degree of R(x) less than degree of G(x)
D) R(x) is always a constant
Answer: C
Explanation: Degree of remainder must always be less than the degree of divisor.
Q. A polynomial with degree zero is called a:
A) Linear polynomial
B) Constant polynomial
C) Quadratic polynomial
D) Zero polynomial
Answer: B
Explanation: A non-zero polynomial of degree zero is called a constant polynomial.
Q. If the zeros of the quadratic polynomial x² + (a+1)x + b are two and minus three, then what are the values of a and b?
A) a = 0, b = -6
B) a = -7, b = -1
C) a = 2, b = -6
D) a = 1, b = 6
Answer: A
Explanation: Sum of zeros = -1 and product = -6. Comparing coefficients gives a = 0 and b = -6.
Q. A cubic polynomial can have a maximum of how many real zeros?
A) 1
B) 2
C) 3
D) 4
Answer: C
Explanation: A polynomial can have at most as many real zeros as its degree.
Q. Which of the following is NOT a polynomial?
A) 5
B) 3x² - 2x + 1
C) y + 1/y
D) z² - 8
Answer: C
Explanation: Variable in denominator makes the expression not a polynomial.
Q. Find the zeros of the polynomial P(x) = x² - 9.
A) 3 only
B) -3 only
C) 3, -3
D) 9
Answer: C
Explanation: x² - 9 = (x - 3)(x + 3), so zeros are 3 and -3.
Understanding Polynomials in Simple Language
A polynomial is an algebraic expression made using variables, constants, and powers. The powers of variables in a polynomial are always non-negative integers.
Examples:
3x + 5
x² + 7x + 10
2x³ − 5x² + x − 4
Polynomials are mainly classified according to their degree.
| Type of Polynomial | Degree |
|---|---|
| Linear Polynomial | 1 |
| Quadratic Polynomial | 2 |
| Cubic Polynomial | 3 |
Questions based on identifying polynomial degree are common in Polynomial multiple choice questions because they test basic concept clarity.
Common Mistakes Students Make in Polynomial MCQs
Even strong students sometimes lose easy marks because of small errors.
Some common mistakes are:
- Ignoring negative signs
- Writing incorrect degree of polynomial
- Confusing factors with zeroes
- Wrong substitution while verifying roots
- Graph interpretation mistakes
- Incorrect formula application
- Forgetting coefficient relationships
Another common issue is solving too quickly without checking calculations properly.
Important Instructions for Solving Polynomials Class 10 MCQs
Read each question carefully before selecting the answer.
- Choose the most appropriate option from the given choices.
- Focus on concepts like zeroes of polynomials, graphs, and coefficient relationships while solving.
- Avoid calculation mistakes and check signs properly in algebraic expressions.
- Try to solve the questions without guessing to improve conceptual understanding and accuracy.
- Practice regularly to improve speed for school exams and CBSE board preparation.
Why Polynomials Chapter is Important for Boards
The Polynomials chapter builds the base for higher algebra concepts. Students who understand this chapter properly usually find later algebra topics easier.
This chapter is important because:
- Questions are frequently asked in exams
- Concepts are connected with algebra fundamentals
- Formula-based objective questions are common
- Graph interpretation skills are tested
- Competency-oriented questions are increasing
Practicing Polynomials Class 10 MCQs with answers helps students prepare more effectively for school exams and board assessments.
Revision Notes for Polynomials
Before exams, students should quickly revise these important points:
- Degree of polynomial depends on highest power
- Zeroes make polynomial equal to zero
- Graph intersects x-axis at zeroes
- Sum of zeroes = −b/a
- Product of zeroes = c/a
- Division Algorithm is very important
- Sign mistakes are very common in objective questions
Short revision regularly improves retention and speed during exams.
Conclusion
Practicing Polynomials Class 10 MCQs with answers is one of the best ways to strengthen algebraic concepts and improve exam performance. The chapter may look formula-based initially, but actual questions often test logic, graph understanding, and conceptual clarity together. Students preparing seriously for CBSE exams should focus on regular practice, careful formula revision, and understanding how polynomial expressions behave mathematically.