1. Introduction

A polynomial is an expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

2. What is a Polynomial?

A polynomial in one variable is an expression of the form:

a_0 + a_1x + a_2x^2 + ... + a_nx^n

Where x is the variable, a_0, a_1, ..., a_n are coefficients, and a_n ≠ 0.

3. Terms Related to Polynomials

• Term: Part of the expression separated by + or -.
• Coefficient: Numerical factor of a term.
• Degree: Highest power of the variable.
• Constant Polynomial: No variable part.
• Zero Polynomial: Polynomial with value 0.

4. Types of Polynomials Based on Number of Terms

• Monomial: One term
• Binomial: Two terms
• Trinomial: Three terms
• Polynomial: One or more terms

5. Types of Polynomials Based on Degree

Degree	Name	Example
0	Constant Polynomial	7, -3
1	Linear Polynomial	3x + 1
2	Quadratic Polynomial	x² - 4x + 3
3	Cubic Polynomial	x³ + 2x² - 1
n	Polynomial of degree n	xⁿ + ...

6. Algebraic Identities

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• a² - b² = (a + b)(a - b)
• (x + a)(x + b) = x² + (a + b)x + ab
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
• (x + y)³ = x³ + 3x²y + 3xy² + y³
• (x - y)³ = x³ - 3x²y + 3xy² - y³
• x³ + y³ = (x + y)(x² - xy + y²)
• x³ - y³ = (x - y)(x² + xy + y²)

7. Zero of a Polynomial

If p(a) = 0, then a is a zero of the polynomial p(x).

8. Value of a Polynomial

Substitute the variable with the given value to evaluate.

9. Addition and Subtraction of Polynomials

Combine like terms.

10. Multiplication of Polynomials

• Monomial × Monomial
• Monomial × Polynomial
• Binomial × Binomial (Use identities)
• Polynomial × Polynomial

11. Division of Polynomials

Involves long division (basic understanding is enough).

12. Factorization of Polynomials

• Taking out common factors
• Using identities

13. Remainder Theorem

When p(x) is divided by x - a, the remainder is p(a).

14. Factor Theorem

If p(a) = 0, then (x - a) is a factor of p(x).

15. Graphical Meaning of Zeros

Zeros correspond to the x-intercepts of the graph of the polynomial.

16. Applications in Real Life

Used in various fields like physics, engineering, computer science, economics, etc.

17. Summary

• Polynomial: sum of terms with whole number exponents
• Types: monomial, binomial, trinomial
• Operations: add, subtract, multiply
• Identities and factorization are essential tools
• Zeros help solve equations

18. Practice Questions

1. Find the degree of the polynomial:
   • a) 4x³ + 5x² - 7
   • b) 3y⁵ - y + 9
2. Factor using identities:
   • a) x² - 16
   • b) x² + 7x + 10
3. Evaluate p(x) = x² - x + 1 at x = -2
4. Use Remainder Theorem to find remainder of p(x) = x² + 4x + 3 when divided by x + 1
5. Check whether x + 3 is a factor of x² + 5x + 6 using Factor Theorem