Page 1: Introduction
Polynomial: expression with variables, coefficients, non-negative integer exponents.
General form: p(x) = a_n x^n + ... + a_0
This chapter covers zeroes, relationship with coefficients, and division algorithm.
Page 2: Types of Polynomials
- By degree: linear (1), quadratic (2), cubic (3), biquadratic (4)
- By terms: monomial, binomial, trinomial
Constant polynomial degree 0, zero polynomial undefined.
Page 3: Zeroes of Polynomial
α is zero if p(α) = 0.
Polynomial of degree n has at most n zeroes.
Graph touches/cuts x-axis at zeroes.
Page 4: Relationship - Quadratic Polynomial
Sum of zeroes = -b/a
Product of zeroes = c/a
Page 5: Relationship - Cubic Polynomial
Sum of zeroes = -b/a
Sum of products two at a time = c/a
Product of zeroes = -d/a
Page 6: Forming Polynomial from Zeroes
If α, β zeroes → (x - α)(x - β) = x² - (α+β)x + αβ
For three zeroes similarly.
Page 7: Division Algorithm
degree r(x) < degree g(x)
Used to find zeroes or factor.
Page 8: Finding Zeroes Using Division
Possible rational zeroes = factors of constant / factors of leading.
Use factor theorem to check.
Page 9: Key Theorems Summary
- Remainder theorem
- Factor theorem
- Relationship between zeroes and coefficients
- Division algorithm
Page 10: Practice Questions - Easy (1-10)
- Degree of quadratic.
- Sum of zeroes for x² - 5x + 6.
- Product of zeroes for 2x² + 4x + 2.
- Form quadratic with zeroes 3,4.
- Number of zeroes for cubic.
- Zero of linear polynomial.
- Verify sum/product.
- Possible rational zeroes of x² - 3x + 2.
- Remainder theorem statement.
- Constant polynomial zeroes.
Page 11: Practice Questions - Medium (11-20)
- Find k if sum of zeroes 5, polynomial x² + kx + 6.
- Zeroes of x³ - 6x² + 11x - 6.
- Form cubic with zeroes 1,2,3.
- Divide x³ + x² + x + 1 by x + 1.
- Find zeroes using factorisation.
- Verify relationship for cubic.
- Find polynomial with zeroes -2, 3.
- Division algorithm example.
- Find k for given zeroes.
- Rational root theorem application.
Page 12: Practice Questions - Hard (21-30)
- If zeroes α, β, γ and α+β+γ=0, find αβγ.
- Form polynomial with given sum and product.
- Find all zeroes if one given.
- Prove using division algorithm.
- Advanced factorisation.
- Relationship for general polynomial.
- Graph and zeroes connection.
- Find k for repeated zero.
- Combined with irrational zeroes.
- Board level proof question.
Page 13: Important Relationships Table
Linear, quadratic, cubic.
Page 14: Common Mistakes
- Wrong sign in sum of zeroes
- Forgetting -d/a in cubic product
- Wrong possible rational zeroes
- Missing factor theorem check
- Confusing degree and number of zeroes
Page 15: Previous Year Questions
Find zeroes, form polynomial, verify relationship, division.
Page 16: Exam Tips
- Memorise sum and product formulas
- Show division steps clearly
- Verify by substitution
- List possible rational zeroes systematically
- Write polynomial in standard form
Page 17: Quick Revision Sheet
All formulas, theorems, relationships.
Page 18: Final Motivation
Chapter 2 complete! Polynomials is high-scoring.
Master zeroes and coefficients relationship.
Class 10 Maths rolling strong 🦖
Page 19: Extra Numericals
More solved problems.
Page 20: Zeroes and Graph
How graph crosses x-axis.
Page 21: Division Algorithm Steps
Detailed examples.
Page 22: Rational Root Theorem
Possible zeroes list.
Page 23: Forming Polynomial
From zeroes method.
Page 24: Relationship Proof Hint
Vieta’s formulas.
Page 25: Board Pattern Questions
Common types.
Page 26: Extra Practice
More questions.
Page 27: Thank You & Copyright
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