Mathematics Notes for Class 11 Polynomials

Polynomials

Welcome to the chapter on Polynomials for Class 11. In this chapter, you will learn about polynomials, their types, operations, factorization, and applications. By the end of this chapter, you will be able to identify, manipulate, and solve problems involving polynomials.

Key Concepts

Polynomial: An algebraic expression consisting of variables and coefficients, involving only non-negative integer powers of the variable.
Degree: The highest power of the variable in a polynomial.
Zero/Root: A value of the variable that makes the polynomial equal to zero.
Factor: An expression that divides the polynomial exactly.

Types of Polynomials

Monomial: A polynomial with one term (e.g., 3x).
Binomial: A polynomial with two terms (e.g., x + 2).
Trinomial: A polynomial with three terms (e.g., x² + 2x + 1).
General Polynomial: An expression with one or more terms (e.g., 2x³ - 4x + 7).

Standard Form of a Polynomial

A polynomial is written in standard form when its terms are arranged in descending order of the powers of the variable.

Example: 2x³ - 5x² + 3x - 7

Operations on Polynomials

Addition: Add like terms.
Subtraction: Subtract like terms.
Multiplication: Multiply each term in one polynomial by each term in the other.
Division: Divide the polynomial by a monomial or another polynomial (using long division or synthetic division).

Factorization of Polynomials

Taking out common factors
Factorization by grouping
Using identities (e.g., a² - b² = (a - b)(a + b))
Splitting the middle term (for quadratics)

Zeros of a Polynomial

The zero or root of a polynomial is the value of the variable for which the polynomial becomes zero. For a quadratic polynomial ax² + bx + c, the zeros can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Applications of Polynomials

Solving equations in algebra
Modeling real-life situations (e.g., area, profit, speed)
Graphing curves and understanding their properties

Practice Questions

Write the standard form of the polynomial: 4 + 3x² - 2x.
Add: (2x² + 3x + 1) and (x² - x + 4).
Factorize: x² - 9.
Find the zeros of the polynomial x² - 5x + 6.
If (x - 2) is a factor of x² - 4x + k, find the value of k.

Challenge Yourself

Divide (2x³ + 3x² - x + 5) by (x + 1).
If α and β are the zeros of x² + px + q, show that α + β = -p and αβ = q.

Did You Know?

The word "polynomial" comes from the Greek words "poly" (many) and "nomial" (term).
Polynomials are used in computer graphics, engineering, and science to model curves and solve problems.

Glossary

Polynomial: An expression with one or more terms, each term having a variable raised to a non-negative integer power.
Degree: The highest exponent of the variable in a polynomial.
Zero/Root: The value of the variable that makes the polynomial zero.
Factorization: Writing a polynomial as a product of its factors.

Answers to Practice Questions

3x² - 2x + 4
(2x² + x²) + (3x - x) + (1 + 4) = 3x² + 2x + 5
(x - 3)(x + 3)
x² - 5x + 6 = 0 ⇒ (x - 2)(x - 3) ⇒ Zeros are x = 2, 3
Put x = 2: (2)² - 4×2 + k = 0 ⇒ 4 - 8 + k = 0 ⇒ k = 4

Practice working with polynomials to master algebra and solve real-world problems!

Mathematics Class 11 - Polynomials Notes