Page 1: Introduction to Polynomials
Polynomials are algebraic expressions with variables, coefficients, and non-negative integer exponents.
General form: p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0
They are building blocks for advanced algebra and appear in real-life applications like physics and economics.
This chapter covers definitions, types, zeroes, remainder theorem, factor theorem, factorization, and algebraic identities.
Master these to score full marks in board exams!
Page 2: Polynomials in One Variable
A polynomial in one variable (usually x) has only one variable.
Examples: 3x² + 2x + 1 (polynomial), x² + 1/√x (not, due to negative/fractional exponent).
Terms: Each part like 3x² (coefficient 3, variable x²).
Constant term: No variable (e.g., 5).
Like/unlike terms: Same variable powers.
Page 3: Degree of a Polynomial
Degree: Highest power of variable with non-zero coefficient.
Examples:
- 5x³ + 2x² + x → degree 3
- 7x + 4 → degree 1
- 9 → degree 0 (constant)
- Zero polynomial → degree undefined or -∞
Degree determines graph shape and roots.
Page 4: Types of Polynomials by Degree
- Constant: Degree 0 (e.g., 10)
- Linear: Degree 1 (e.g., 2x + 3)
- Quadratic: Degree 2 (e.g., x² + 4x + 4)
- Cubic: Degree 3 (e.g., x³ - x² + 2)
- Biquadratic: Degree 4
Linear has 1 root, quadratic up to 2, cubic up to 3.
Page 5: Types by Number of Terms
- Monomial: 1 term (e.g., 5x³)
- Binomial: 2 terms (e.g., x² + 1)
- Trinomial: 3 terms (e.g., x² + 2x + 1)
Page 6: Zeroes of a Polynomial
Zero/root/α: Value where p(α) = 0.
Linear: 1 zero.
Quadratic: Up to 2 (real/distinct/repeated).
Cubic: Up to 3.
Polynomial degree n has at most n real zeroes.
Page 7: Remainder Theorem
If p(x) divided by (x - a), remainder = p(a).
p(x) = (x - a) q(x) + r (r constant).
Put x = a: p(a) = r.
p(-1) = 2(1) - 5 + 3 = 0 → remainder 0.
Page 8: Factor Theorem
(x - a) is factor of p(x) ⇔ p(a) = 0.
Converse of remainder theorem when remainder 0.
p(2) = 8 - 24 + 22 - 6 = 0 → yes.
Page 9: Factorisation Using Theorems
Find possible factors (rational roots ± factors of constant/leading).
Use factor theorem → divide → repeat.
Possible: ±1, ±2.
p(1)=0 → (x-1) factor → etc.
Page 10: Algebraic Identities Overview
Identities hold for all values → key for factorization.
Main ones:
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- (a + b)(a - b) = a² - b²
Page 11: More Identities
- (x + a)(x + b) = x² + (a+b)x + ab
- (x + a)(x - b) = x² + (a-b)x - ab
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)
Page 12: Using Identities for Factorisation
x² - 9 = (x + 3)(x - 3)
Page 13: Key Formulas Summary
- Remainder Theorem: p(a) = remainder when divided by (x - a)
- Factor Theorem: p(a)=0 ⇒ (x - a) factor
- (a + b)² = a² + 2ab + b²
- a² - b² = (a + b)(a - b)
- a³ + b³ = (a + b)(a² - ab + b²)
Page 14: Practice Questions - Easy (1-10)
- Find degree of 4x⁴ - 3x + 2.
- Classify 5x² + 1 (type by degree/terms).
- Find zeroes of x - 5.
- Is x+1 factor of x² + 2x + 1? Check.
- Expand (3x + 2)².
- Factor x² - 16.
- Find remainder: (x³ + 1) ÷ (x + 1).
- Verify (a + b)² identity for a=2, b=3.
- Write binomial degree 50.
- Zeroes of quadratic?
Page 15: Practice Questions - Medium (11-20)
- Factor x² + 5x + 6.
- Use remainder theorem on x⁴ - 1 ÷ (x - 1).
- Factor 9x² - 25.
- Simplify (2x + 3)(2x - 3).
- Find zeroes of x² - 4x + 4.
- Factor x³ + 8.
- Verify factor theorem for x³ - 6x² + 11x - 6 and x=1.
- Expand (x + y + z)² (advanced).
- Factor 4x² + 12x + 9.
- Remainder of 3x² + 7x + 2 ÷ (x + 2).
Page 16: Practice Questions - Hard (21-30)
- Factor x³ - 23x² + 142x - 120.
- Prove (x - 2) factor of x⁴ - 5x² + 4? Wait, check.
- Factor completely 27x³ - 8.
- Use identities: x⁶ - y⁶.
- Find k if (x + k) factor of x³ + kx² - 2kx + 4.
- Divide x⁴ + x² + 1 by x² + x + 1.
- Factor x⁴ + 4.
- Zeroes of cubic with one known zero.
- Verify a³ + b³ + c³ - 3abc = (a+b+c)(... ) if a+b+c=0.
- Complex factorisation.
Page 17: NCERT Exercise 2.1 Type
Focus on identifying polynomials, degree, types.
Common: Which are polynomials? State reasons.
Page 18: NCERT Exercise 2.2 Type
Zeroes of polynomials, verification.
Find zeroes and verify sum/product (for quadratic).
Page 19: NCERT Exercise 2.3 Type
Remainder and factor theorem applications.
Find remainders, check factors.
Page 20: NCERT Exercise 2.4 Type
Factorisation using identities and theorems.
Heavy on algebraic identities.
Page 21: Important Theorems Recap
- Remainder Theorem
- Factor Theorem
- Maximum n zeroes for degree n
- All standard algebraic identities
Page 22: Common Mistakes to Avoid
- Forgetting constant term in expansion.
- Wrong conjugate in difference of squares.
- Miscalculating p(a) in theorems.
- Thinking non-integer exponents allowed.
- Forgetting to check all possible factors.
Page 23: Previous Year Board Questions
Typical: Factorise using identity (3 marks).
Prove factor using theorem (4 marks).
Find zeroes and verify (2 marks).
Remainder questions (1-2 marks).
Page 24: Exam Strategy Tips
- Memorise all identities perfectly.
- Show steps in factorisation.
- Check p(a)=0 clearly.
- Practice all NCERT exercises twice.
- Revise theorems with proofs.
Page 25: Quick Revision Formula Sheet
- (a + b)² = a² + 2ab + b²
- a² - b² = (a + b)(a - b)
- a³ + b³ = (a + b)(a² - ab + b²)
- Remainder = p(a)
- p(a)=0 ⇒ factor
Master these → ace Polynomials!
Page 26: Final Motivation
You've completed this 27-page Polynomials guide!
Polynomials are super important for Class 10 and beyond.
Practice daily, solve extras, stay confident.
Board Buddy got your back 🦖
Crush those exams!
Page 27: Thank You & Copyright
Made with ❤️ by Grok exclusively for
BOARD BUDDY
Class 9 & 10 Students
© 2025 - All Rights Reserved for Educational Use