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Class 10th Polynomial Notes, Study notes of Mathematics

Mathematics

It contains full detailed notes and contain a lot of practice for the chapter which will help students to learn effectively

Typology: Study notes

2025/2026

Available from 03/28/2026

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📘 POLYNOMIALS – COACHING INSTITUTE NOTES
(CLASS 10)
Polynomials form the foundation of algebra. This chapter
focuses on:
Understanding polynomial expressions
Finding zeros
Relationships between zeros and coefficients
Division algorithm and factor theorem
📘 1. POLYNOMIAL – DEFINITION
A polynomial in one variable x is of the form:
p(x) = a xⁿ + a xⁿ ¹ + ... + a x + aₙ ₙ₋₁ ⁻ ₁ ₀
Where:
a ≠ 0ₙ
n is a non-negative integer
📘 2. TYPES OF POLYNOMIALS
✔ Based on Number of Terms
Type Example
Monomial 5x
Binomial x + 2
Trinomial x² + 3x + 2
pf3
pf4
pf5
pf8
pf9

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📘 POLYNOMIALS – COACHING INSTITUTE NOTES

(CLASS 10)

Polynomials form the foundation of algebra. This chapter focuses on:  Understanding polynomial expressions  Finding zeros  Relationships between zeros and coefficients  Division algorithm and factor theorem 📘 1. POLYNOMIAL – DEFINITION A polynomial in one variable x is of the form: p(x) = a xⁿ + aₙ ₙ₋₁xⁿ ¹ + ... + a x + a ⁻ ₁ ₀ Where:  a ₙ≠ 0  n is a non-negative integer 📘 2. TYPES OF POLYNOMIALS ✔ Based on Number of Terms Type Example Monomial 5x Binomial x + 2 Trinomial x² + 3x + 2

✔ Based on Degree Degree Name Example 0 Constant 5 1 Linear x + 2 2 Quadratic x² + 3x + 1 3 Cubic x³ + x 📘 3. DEGREE OF POLYNOMIAL The highest power of x in the polynomial. Example:  5x³ + 2x → Degree = 3  7 → Degree = 0 📘 4. VALUE OF A POLYNOMIAL Substitute value of x. Example: p(x) = x² + 2x + 1 p(2) = 4 + 4 + 1 = 9 📘 5. ZEROS OF A POLYNOMIAL A value α is a zero if: p(α) = 0

📘 9. DIVISION ALGORITHM

p(x) = g(x)·q(x) + r(x) Condition: Degree of r(x) < Degree of g(x) 📘 10. REMAINDER THEOREM Remainder when p(x) is divided by (x – a) = p(a) 📘 11. FACTOR THEOREM If p(a) = 0 → (x – a) is a factor 📘 12. IMPORTANT IDENTITIES  (a + b)² = a² + 2ab + b²  (a – b)² = a² – 2ab + b²  a² – b² = (a – b)(a + b)  (a + b)³ = a³ + b³ + 3ab(a + b)  (a – b)³ = a³ – b³ – 3ab(a – b) 📘 SOLVED EXAMPLES Example 1: Find zero of x + 5

Solution: x + 5 = 0 x = – Example 2: Find sum and product of zeros of 2x² + 5x + 3 Solution: a = 2, b = 5, c = 3 Sum = –b/a = –5/ Product = c/a = 3/ Example 3: Check if (x – 2) is factor of x² – 4 p(2) = 4 – 4 = 0 Hence, factor ✔ ⚡ COMMON MISTAKES ❌ Forgetting negative sign in –b/a ❌ Writing wrong degree ❌ Not simplifying completely 📘 EXAM STRATEGY

  1. Find zeros of x² + 5x + 6
  2. Find value of p(–1) for p(x)=x²+2x+
  3. Find degree of 7x³ + 2x² + x
  4. Check if (x+2) is factor of x² + 3x + 2
  5. Find remainder when x² + x + 1 is divided by (x+1)
  6. Find sum of zeros of x² – 4x + 7
  7. Find product of zeros of 3x² + x – 2
  8. Form polynomial with zeros –1, 4
  9. Find value of p(0) for p(x)=2x²+ 📘 LEVEL 2 (Exam Level)
  10. Find zeros of 2x² – 7x + 3
  11. Verify relation between zeros & coefficients for x² – 3x + 2
  12. Find polynomial with sum = 5, product = 6
  13. Check if (x–3) is factor of x³ – 27
  14. Find remainder when x³ – 2x² + 4 is divided by (x–
  1. Find value of k if (x–1) is factor of x² + kx – 2
  2. Find zeros of x² – 2√5x + 5
  3. Find polynomial with zeros 1/2 and –
  4. If α,β are zeros of x² – 6x + 8, find α² + β²
  5. Find sum of cubes of zeros of x² – 3x + 1 📘 LEVEL 3 (HIGH VALUE / SELLING POINT)
  6. If α,β are zeros of x² – 5x + 6, find (α/β + β/α)
  7. Find polynomial whose zeros are squares of zeros of x² – 3x + 2
  8. If α,β are zeros of ax² + bx + c, prove α+β = –b/a
  1. Find k such that x² + kx + 9 has equal zeros
  2. Find nature of zeros of x² – 4x + 5
  3. Find cubic polynomial with zeros 1,2,
  4. If α,β are zeros, find α³ + β³ for x² – x –
  5. Find polynomial whose zeros are reciprocal of zeros of x² – 5x + 6
  6. If α+β=7, αβ=10, form polynomial
  7. Find remainder when x⁴ – 1 is divided by (x–1) 📘 CASE-BASED QUESTIONS (CBSE PATTERN)
  8. A polynomial p(x)=x²–5x+6 represents profit. Find when profit = 0
  9. Graph of polynomial cuts x-axis at 2 and 3. Form polynomial
  10. If graph touches x-axis once, what can you say about zeros?
  11. Given polynomial x²+kx+9 has equal roots, find k
  12. If p(2)=0 and p(x)=x²+ax+2, find a ⚡ ASSERTION-REASON
  13. Assertion: Degree of polynomial can be negative Reason: Power of variable can be negative
  14. Assertion: Every linear polynomial has one zero Reason: Graph cuts x-axis once
  15. Assertion: If p(a)=0, then (x–a) is factor Reason: Factor theorem 📘 HOT MIXED QUESTIONS (FOR SPEED PRACTICE)