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Mathematics Class 10 - Polynomials Notes

Comprehensive study notes for Class 10 - Polynomials olympiad preparation

Polynomials

Welcome to the chapter on Polynomials for Class 10. In this chapter, you will learn what polynomials are, their types, how to perform operations on them, and how to find their zeroes. By the end of this chapter, you will be able to solve problems involving polynomials and understand their applications in algebra.

Key Concepts

  • Polynomial: An algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents.
  • Degree of a Polynomial: The highest power of the variable in the polynomial.
  • Zero of a Polynomial: A value of the variable for which the polynomial becomes zero.

Types of Polynomials

  • Constant Polynomial: A polynomial of degree 0 (e.g., 5).
  • Linear Polynomial: A polynomial of degree 1 (e.g., 2x + 3).
  • Quadratic Polynomial: A polynomial of degree 2 (e.g., x² + 2x + 1).
  • Cubic Polynomial: A polynomial of degree 3 (e.g., x³ - x² + 2).

Standard Form of a Polynomial

A polynomial in one variable x is written as:
P(x) = anxⁿ + an-1xⁿ⁻¹ + ... + a₁x + a₀
where an, an-1, ..., a₀ are real numbers and n is a non-negative integer.

Zeroes of a Polynomial

The zero of a polynomial P(x) is the value of x for which P(x) = 0.
Example: For P(x) = x² - 4, the zeroes are x = 2 and x = -2.

Operations on Polynomials

  • Addition: Add the like terms of two polynomials.
  • Subtraction: Subtract the like terms of two polynomials.
  • Multiplication: Multiply each term of one polynomial by each term of the other.
  • Division: Divide one polynomial by another using long division.

Factorization of Polynomials

Factorization means writing a polynomial as a product of its factors.
Example: x² - 5x + 6 = (x - 2)(x - 3)

Remainder and Factor Theorems

  • Remainder Theorem: If a polynomial P(x) is divided by (x - a), the remainder is P(a).
  • Factor Theorem: (x - a) is a factor of P(x) if and only if P(a) = 0.

Practice Questions

  1. Find the degree of the polynomial: 3x⁴ + 2x² - 7.
  2. If P(x) = x² - 9, find its zeroes.
  3. Add: (2x² + 3x + 1) and (x² - x + 4).
  4. Factorize: x² + 5x + 6.
  5. If P(x) = x³ - 2x² + x - 2, find the remainder when divided by (x - 2).

Challenge Yourself

  • Divide x³ + 2x² - x - 2 by x + 1 using long division.
  • Show that (x - 3) is a factor of x² - 6x + 9 using the factor theorem.

Did You Know?

  • The word "polynomial" comes from the Greek words "poly" (many) and "nomial" (term).
  • Polynomials are used in physics, engineering, economics, and many other fields.

Glossary

  • Polynomial: An expression with variables, coefficients, and non-negative integer exponents.
  • Degree: The highest exponent of the variable in a polynomial.
  • Zero: A value of the variable that makes the polynomial equal to zero.
  • Factorization: Writing a polynomial as a product of its factors.

Answers to Practice Questions

  1. 4
  2. x = 3, x = -3
  3. 2x² + x + 5
  4. (x + 2)(x + 3)
  5. P(2) = 8 - 8 + 2 - 2 = 0 (remainder is 0)

Practice solving polynomial problems to strengthen your algebra skills!

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