MP Board 10th Maths: Polynomials (बहुपद) Ch 2 — Notes & PYQs
Chapter 2: Polynomials (बहुपद) — Polynomials form a crucial foundation in algebra for Class 10 MP Board Mathematics. This chapter carries approximately 8–10 marks in the board exam, typically featuring 2 short-answer questions (2–3 marks each), 1 long-answer question (4–5 marks), and 1–2 objective-type questions. You will learn about polynomial degrees, zeros (roots), the relationship between coefficients and zeros, and division algorithms. Mastering this chapter not only helps in board exams but also builds essential algebra skills for higher studies in Class 11 and 12.
📑 Table of Contents
- 1. Introduction to Polynomials
- 2. Degree of a Polynomial
- 3. Types of Polynomials (Based on Degree)
- 4. Zeros of a Polynomial
- 5. Relationship Between Zeros and Coefficients
- 6. Division Algorithm for Polynomials
- 7. Geometrical Meaning of Zeros
- 8. Important Formulae and Identities
- 9. Practice Questions with Solutions
- 10. Previous Year Questions (2017–2026)
- 11. Frequently Asked Questions
🔢 1. Introduction to Polynomials
A polynomial is an algebraic expression consisting of variables, constants, and exponents combined using addition, subtraction, multiplication, and non-negative integer powers. In simpler terms, any expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ where the exponents (n, n−1, …) are whole numbers, is called a polynomial in one variable x.
The word “polynomial” comes from Greek — “poly” means many and “nomial” means terms. So a polynomial can have many terms. The coefficients (aₙ, aₙ₋₁, … a₀) are real numbers, and aₙ ≠ 0. The highest exponent in the polynomial is called its degree.
✅ 3x² + 2x − 5 (degree 2)
✅ 4x³ − 7x + 9 (degree 3)
✅ −2x + 6 (degree 1, linear polynomial)
✅ 7 (degree 0, constant polynomial)
NOT Polynomials:
❌ 5x⁻² + 3 (negative exponent)
❌ √x + 4 (fractional exponent ½)
❌ 3/x + 2 (variable in denominator)
📐 2. Degree of a Polynomial
The degree of a polynomial is the highest exponent of the variable in the polynomial. It tells us the highest power of x that appears in the expression. The degree determines the polynomial’s name and behaviour on a graph.
Finding Degree of a Multivariable Polynomial
When a polynomial has more than one variable (e.g., 3x²y + 2xy³ − 5), the degree is the highest sum of exponents in any single term. In 3x²y, exponents are 2 + 1 = 3. In 2xy³, exponents are 1 + 3 = 4. So the degree is 4.
📊 3. Types of Polynomials (Based on Degree)
3.1 Linear Polynomials (Degree 1)
Form: p(x) = ax + b, where a ≠ 0. Graph is a straight line. Has exactly ONE zero (root). Example: 2x − 6 has zero at x = 3.
3.2 Quadratic Polynomials (Degree 2)
Form: p(x) = ax² + bx + c, where a ≠ 0. Graph is a parabola (U-shaped curve). Has at most TWO zeros. Examples: x² − 5x + 6, 2x² + 3x − 2.
3.3 Cubic Polynomials (Degree 3)
Form: p(x) = ax³ + bx² + cx + d, where a ≠ 0. Has at most THREE zeros. Example: x³ − 6x² + 11x − 6 has zeros at x = 1, 2, 3.
3.4 Classification by Number of Terms
🎯 4. Zeros of a Polynomial
A zero (or root) of a polynomial p(x) is a value of x for which p(x) = 0. In other words, if p(α) = 0, then α is called a zero of the polynomial p(x). For example, if p(x) = x² − 4, then p(2) = 4 − 4 = 0 and p(−2) = 4 − 4 = 0, so 2 and −2 are zeros of this polynomial.
Key Points about Zeros
- A linear polynomial (ax + b) has exactly one zero: x = −b/a
- A quadratic polynomial (ax² + bx + c) has at most two zeros
- A cubic polynomial (ax³ + bx² + cx + d) has at most three zeros
- An n-degree polynomial has at most n zeros (some may be repeated)
- Zeros are also called roots or solutions of the polynomial equation
How to Find Zeros of a Quadratic Polynomial
Method 1 — Factorisation: Express ax² + bx + c as the product of two linear factors. Example: x² − 7x + 12 = (x − 3)(x − 4). So zeros are 3 and 4.
Method 2 — Quadratic Formula: When factorisation is not easy, use the formula:
x = [−b ± √(b² − 4ac)] / 2a
Where:
D = b² − 4ac is called the Discriminant
D > 0 → Two distinct real zeros
D = 0 → One repeated real zero (equal roots)
D < 0 → No real zeros (complex zeros)
📊 5. Relationship Between Zeros and Coefficients
For a Quadratic Polynomial p(x) = ax² + bx + c
Let α and β be the zeros of the quadratic polynomial ax² + bx + c. Then:
For a Cubic Polynomial p(x) = ax³ + bx² + cx + d
Let α, β, and γ be the zeros. Then:
➗ 6. Division Algorithm for Polynomials
The Division Algorithm states: If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that:
Where degree of r(x) < degree of g(x)
Step-by-Step Division Example
Example: Divide p(x) = 3x³ + x² + 2x + 5 by g(x) = x² + 2x + 1
- Step 1: Divide the highest degree term. 3x³ ÷ x² = 3x (first term of quotient)
- Step 2: Multiply: 3x × (x² + 2x + 1) = 3x³ + 6x² + 3x
- Step 3: Subtract: (3x³ + x² + 2x + 5) − (3x³ + 6x² + 3x) = −5x² − x + 5
- Step 4: Repeat: −5x² ÷ x² = −5 (second term of quotient)
- Step 5: Multiply: −5 × (x² + 2x + 1) = −5x² − 10x − 5
- Step 6: Subtract: (−5x² − x + 5) − (−5x² − 10x − 5) = 9x + 10
Result: Quotient q(x) = 3x − 5, Remainder r(x) = 9x + 10
Verification: (x² + 2x + 1)(3x − 5) + (9x + 10) = 3x³ + x² + 2x + 5 ✓
📘 Key Fact: The Division Algorithm also helps in finding zeros of a polynomial when one zero is known. If we know one zero of a cubic polynomial, we can divide the cubic by (x − known zero) to get a quadratic, then solve that quadratic to find the other two zeros.
📈 7. Geometrical Meaning of Zeros
The zero of a polynomial is the x-coordinate where the graph of the polynomial crosses the x-axis (where y = 0). This visual understanding is extremely important for board exams.
Linear Polynomial: y = ax + b
Graph is a straight line. It intersects the x-axis at exactly ONE point. That point’s x-coordinate is the zero. For y = 2x − 6, the line crosses x-axis at (3, 0).
Quadratic Polynomial: y = ax² + bx + c
Graph is a parabola (U-shaped). The number of times it touches/crosses the x-axis gives the number of real zeros:
- Two distinct points → D > 0 → Two distinct real zeros
- One point (touches) → D = 0 → One repeated zero
- No intersection → D < 0 → No real zeros
If a > 0, parabola opens upward. If a < 0, parabola opens downward.
Cubic Polynomial: y = ax³ + bx² + cx + d
Graph is an S-shaped curve that can cross the x-axis up to 3 times. The maximum number of real zeros equals the degree (3), but some zeros may be complex numbers that don’t appear on the real plane.
📝 8. Important Formulae and Identities
✏️ 9. Practice Questions with Solutions
Q1. Find the zeros of the quadratic polynomial x² + 7x + 10
Solution: x² + 7x + 10 = x² + 2x + 5x + 10 = x(x + 2) + 5(x + 2) = (x + 2)(x + 5)
Therefore, zeros are x = −2 and x = −5.
Q2. Find a quadratic polynomial whose zeros are 4 and −3.
Solution: Using the formula p(x) = x² − (sum)x + (product)
Sum = 4 + (−3) = 1
Product = 4 × (−3) = −12
Therefore, p(x) = x² − (1)x + (−12) = x² − x − 12
Q3. If α and β are the zeros of x² − 3x + 2, find α²β + αβ².
Solution: For x² − 3x + 2, sum α + β = 3, product αβ = 2
α²β + αβ² = αβ(α + β) = 2 × 3 = 6
Q4. Find the value of k such that the polynomial x² − (k + 3)x + 2k has sum of zeros equal to 5.
Solution: For x² − (k + 3)x + 2k, a = 1, b = −(k + 3)
Sum of zeros = −b/a = k + 3
Given: k + 3 = 5, so k = 2
Q5. Divide 2x³ + 3x² − 11x − 6 by x − 2 and find the quotient and remainder.
Solution: Using synthetic/long division:
2x³ ÷ x = 2x², then multiply → 2x³ − 4x², subtract → 7x² − 11x
7x² ÷ x = 7x, multiply → 7x² − 14x, subtract → 3x − 6
3x ÷ x = 3, multiply → 3x − 6, subtract → 0
Quotient = 2x² + 7x + 3, Remainder = 0
So (x − 2) is a factor, and 2 is a zero of the polynomial.
📋 Previous Year Questions (2017–2026)
❓ Frequently Asked Questions
Q1. What is a polynomial in simple words?
A polynomial is an algebraic expression made of variables and constants using only addition, subtraction, and multiplication — no division by a variable, no negative exponents, and no square roots of variables. In Class 10, we mainly study polynomials in one variable like x where the exponents are whole numbers (0, 1, 2, 3, …).
Q2. What is the difference between a zero and a root?
They mean the same thing. A number α is called a zero of polynomial p(x) if p(α) = 0. It is also called a root of the polynomial equation p(x) = 0. For example, 2 and 3 are both zeros and roots of x² − 5x + 6.
Q3. Can a quadratic polynomial have 3 zeros?
No, absolutely not. A quadratic polynomial has degree 2, so it can have at most 2 zeros. This is a fundamental rule of algebra — a degree n polynomial has at most n zeros. A cubic polynomial (degree 3) can have up to 3 zeros.
Q4. What is the discriminant and why is it important?
The discriminant D = b² − 4ac tells us the nature of zeros without actually solving the equation. If D > 0, there are two distinct real zeros. If D = 0, there is exactly one repeated real zero (also called equal roots). If D < 0, there are no real zeros — the zeros are complex numbers. This concept is tested frequently in MP Board objective questions.
Q5. How do I find the polynomial when zeros are given?
This is one of the most common exam questions. If α and β are zeros of a quadratic polynomial, the polynomial is x² − (α + β)x + αβ. If you need a polynomial with leading coefficient other than 1, multiply by k: k[x² − (α + β)x + αβ]. For cubic polynomials with zeros α, β, γ, the formula is: x³ − (α+β+γ)x² + (αβ+βγ+γα)x − αβγ.
Q6. What is the Division Algorithm in polynomials?
The Division Algorithm states that for any two polynomials p(x) and g(x) (with g(x) ≠ 0), there exist unique polynomials q(x) and r(x) such that p(x) = g(x)·q(x) + r(x), where the degree of r(x) is strictly less than the degree of g(x). This is the polynomial version of how we divide numbers — for example, 17 ÷ 5 = 3 remainder 2, which means 17 = 5×3 + 2.
Q7. Can a polynomial have no real zeros?
Yes. If the discriminant D < 0, the quadratic polynomial has no real zeros. This means the parabolic graph of the polynomial never touches or crosses the x-axis. For example, x² + x + 1 has D = 1 − 4 = −3 < 0, so it has no real zeros and its parabola sits entirely above the x-axis.
Q8. How do I verify the relationship between zeros and coefficients?
This is a standard 3-mark question in MP Board. First find the zeros of the polynomial using factorisation or the quadratic formula. Then compute the sum of zeros (α + β) and the product of zeros (α × β). Finally, compare these with −b/a and c/a respectively. For example, for x² − 5x + 6, zeros are 2 and 3. Sum = 5 = −(−5)/1 ✓. Product = 6 = 6/1 ✓.
Q9. How many marks does Polynomials carry in MP Board exam?
The Polynomials chapter carries approximately 8–10 marks in the MP Board Class 10 Mathematics exam. This typically includes 2 objective-type questions (1 mark each), 1 short-answer question worth 2–3 marks, and 1 long-answer question worth 4–5 marks. The most common topics tested are finding zeros, relationship between zeros and coefficients, and the division algorithm.
Q10. What is the difference between monomial, binomial, and trinomial?
These classifications are based on the number of terms, not the degree. A monomial has exactly 1 term (e.g., 5x², 7, −3x). A binomial has 2 terms (e.g., x² − 4, 2x + 1). A trinomial has 3 terms (e.g., x² + 5x + 6). A polynomial with 4 or more terms is simply called a polynomial with no special name.
Q11. What does it mean graphically when a polynomial has equal zeros?
When D = 0, the quadratic polynomial has equal or repeated zeros. Graphically, this means the parabola just touches the x-axis at exactly one point (the vertex of the parabola lies on the x-axis). For example, x² − 2x + 1 = (x − 1)² has repeated zero at x = 1, and its parabola touches the x-axis at (1, 0).
Q12. How can I check if a given expression is a polynomial?
Check these three conditions: (1) All exponents of the variable must be whole numbers (0, 1, 2, 3, …). (2) No variable should appear in the denominator. (3) No variable should appear under a radical (square root, cube root, etc.). If any of these is violated, the expression is not a polynomial. For example, x² + 1/x is not a polynomial because x appears in the denominator.
Q13. What is the zero of the linear polynomial ax + b?
The zero of a linear polynomial ax + b is x = −b/a. This is because setting ax + b = 0 gives ax = −b, therefore x = −b/a. For example, the zero of 2x − 6 is x = 3, since 2(3) − 6 = 0. This formula is very useful for solving linear equations quickly.
Q14. Why do we study polynomials?
Polynomials are the building blocks of algebra and appear everywhere in higher mathematics — from calculus (where polynomial functions are the easiest to differentiate and integrate) to physics (describing motion with quadratic equations) to economics (profit and cost functions). Mastering polynomials in Class 10 sets a strong foundation for all future mathematics.
Q15. What is the best way to prepare for Polynomials in board exams?
Follow this 3-step strategy: First, master the basic concepts — degree, types of polynomials, and the definition of zeros. Second, practice the relationship between zeros and coefficients for both quadratic and cubic polynomials until you can do it quickly. Third, solve at least 15–20 past board exam questions from 2017 to 2026. Focus especially on the division algorithm and questions that ask you to “find the value of k” — these are examiner favourites and appear every year.