Understanding Polynomials

Definition

• Polynomial: A mathematical expression consisting of many terms.
  • Pali (Greek): Means "many."
  • Nomial (Latin): Derived from "gnomon" meaning "name" but refers to terms in mathematics.

Examples of Polynomials

• Polynomials:
  • Example 1: 10x^7 - 9x^2 + 15x^3 + 9
  • Example 2: 9a^2 - 5
  • Example 3: 6 (considered a polynomial)
  • Example 4: 7x^2 - 3
  • Example 5: 7y^2 - 3y + π

Non-Examples of Polynomials

• Not Polynomials:
  • Example 1: 10x^{-7} - 9x^2 + 15x^3 + 9 (negative exponent)
  • Example 2: 9a^{1/2} - 5 (non-integer exponent)
  • Example 3: 9a^a - 5 (variable exponent)

Rules for Polynomials

• Must have non-negative integer exponents.
• A polynomial consists of a finite number of terms.

Terms in Polynomials

• Terms: Individual components of a polynomial.
  • Example: In 10x^7 - 9x^2 + 15x^3 + 9, the terms are:
    • 10x^7 (1st term)
    • -9x^2 (2nd term)
    • 15x^3 (3rd term)
    • 9 (4th term)
• Coefficient: The number multiplying the variable (e.g., in 10x^7, the coefficient is 10).
  • A term can be represented as ax^n where:
    • a is the coefficient.
    • x is the variable.
    • n is a non-negative integer.

Classifications of Polynomials

• Monomial: 1 term (e.g., 6x^0 or 10z^{15}).
• Binomial: 2 terms (e.g., 3y^3 + 5y).
• Trinomial: 3 terms (e.g., 10x^2 - 3x + 4).

Degree of Polynomials

• Degree of a Term: The exponent of the variable in the term.
• Degree of the Polynomial: The highest degree among its terms.
  • Example: 10x^7 - 9x^2 is a 7th degree polynomial.

Leading Term and Coefficient

• Leading Term: The term with the highest degree in a polynomial.
• Leading Coefficient: The coefficient of the leading term.

Standard Form of Polynomials

• Standard Form: Polynomial written in descending order of degree.
  • Example: 10x^7 + 15x^3 - 9x^2 + 9 is in standard form.
• Importance: Helps identify leading term and leading coefficient easily.
  • Leading term: 10x^7 (leading coefficient is 10).

Conclusion

• Understanding the structure and terminology of polynomials is essential for further studies in mathematics.