Polynomials
Chapter 2 | Mathematics
Complete NCERT notes, important formulas, types of polynomials, zeros of polynomials, relationship between zeros and coefficients, division algorithm, and key points for CBSE Class 10 Board Exam 2025-26.
📖 Introduction to Polynomials
Polynomial: An expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ, aₙ₋₁, ..., a₀ are real numbers and aₙ ≠ 0. The highest power of x is called the degree of the polynomial.
📊 Types of Polynomials
🔹 Linear Polynomial
p(x) = ax + b, a ≠ 0
Degree: 1
Example: 2x + 3, 5x - 7
Number of zeros: 1
🔸 Quadratic Polynomial
p(x) = ax² + bx + c, a ≠ 0
Degree: 2
Example: x² - 5x + 6, 2x² + 3x - 1
Number of zeros: 2
🔹 Cubic Polynomial
p(x) = ax³ + bx² + cx + d, a ≠ 0
Degree: 3
Example: x³ - 6x² + 11x - 6
Number of zeros: 3
🔸 Zero Polynomial
p(x) = 0
Degree: Not defined
💡 Note: A polynomial of degree n has at most n zeros (roots).
🔢 Zeros of Polynomials
Zero of a Polynomial: A real number α is called a zero of polynomial p(x) if p(α) = 0.
✅ Example: Linear Polynomial
p(x) = 2x - 6
2x - 6 = 0 → x = 3
∴ Zero of p(x) is 3
✅ Example: Quadratic Polynomial
p(x) = x² - 5x + 6
x² - 5x + 6 = (x - 2)(x - 3)
∴ Zeros are 2 and 3
🔗 Relationship Between Zeros and Coefficients
📐 For Quadratic Polynomial ax² + bx + c
Let α and β be zeros, then:
α + β = -b/a
αβ = c/a
📐 For Cubic Polynomial ax³ + bx² + cx + d
Let α, β, γ be zeros, then:
α + β + γ = -b/a
αβ + βγ + γα = c/a
αβγ = -d/a
🔍 Example: Quadratic
p(x) = x² - 5x + 6
a=1, b=-5, c=6
Zeros: 2 and 3
Sum = 5 = -(-5)/1 = 5 ✓
Product = 6 = 6/1 = 6 ✓
🔍 Example: Cubic
p(x) = x³ - 6x² + 11x - 6
Zeros: 1, 2, 3
Sum = 6 = -(-6)/1 = 6 ✓
Product = 6 = -(-6)/1 = 6 ✓
➗ Division Algorithm for Polynomials
📜 Division Algorithm Statement
If p(x) and g(x) are two polynomials with g(x) ≠ 0, then there exist unique polynomials q(x) and r(x) such that:
p(x) = g(x) × q(x) + r(x), where r(x) = 0 or deg r(x) < deg g(x)
💡 Note: If r(x) = 0, then g(x) is a factor of p(x).
📝 NCERT Solved Examples
Example 1: Find zeros of quadratic polynomial x² + 7x + 10
x² + 7x + 10 = x² + 5x + 2x + 10
= x(x + 5) + 2(x + 5)
= (x + 5)(x + 2)
∴ Zeros are -5 and -2
Example 2: Verify relationship between zeros and coefficients
For p(x) = x² - 3x - 10
Zeros: 5 and -2
Sum = 5 + (-2) = 3 = -(-3)/1 = 3 ✓
Product = 5 × (-2) = -10 = -10/1 = -10 ✓
Example 3: Find quadratic polynomial with sum 1/4 and product -1
Sum = 1/4, Product = -1
Polynomial = x² - (Sum)x + (Product)
= x² - (1/4)x + (-1)
Multiply by 4: 4x² - x - 4
Example 4: Divide x³ - 3x² + 5x - 3 by x² - 2
Quotient = x - 3
Remainder = 7x - 9
Verification: (x² - 2)(x - 3) + (7x - 9) = x³ - 3x² - 2x + 6 + 7x - 9 = x³ - 3x² + 5x - 3 ✓
⭐ Key Points for Board Exam
- Degree of polynomial: Highest power of variable in the polynomial.
- Zero polynomial: p(x) = 0 (degree not defined).
- Number of zeros: A polynomial of degree n has at most n real zeros.
- Graph of linear polynomial is a straight line.
- Graph of quadratic polynomial is a parabola (opens upward if a > 0, downward if a < 0).
- Discriminant (D) for quadratic: D = b² - 4ac
- Nature of roots (zeros): D > 0 → 2 distinct real roots, D = 0 → 1 real root (repeated), D < 0 → no real roots.
- Zero polynomial has infinitely many zeros.
- Division Algorithm is used to find quotient and remainder.