Polynomials

Definition

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• The variables are raised to non-negative integer powers.

Terms

• A term is a single element of a polynomial, consisting of a coefficient and a variable(s) raised to a power.
• Examples: 3x, 2x^2, 5

Degrees

• The degree of a term is the sum of the powers of the variables.
• The degree of a polynomial is the highest degree of its terms.
• Examples:
  • The degree of 3x is 1.
  • The degree of 2x^2 is 2.
  • The degree of 4x^2 + 2x + 1 is 2.

Classification

• Polynomials can be classified based on their degree:
  • Monomials (degree 1): 2x, 3y
  • Binomials (degree 2): x^2 + 3x, 2y^2 - 4y
  • Trinomials (degree 3): x^3 + 2x^2 - 3x, 2y^3 - 4y^2 + 5y
• Polynomials can also be classified based on the number of terms:
  • Monomials (1 term): 2x, 3
  • Binomials (2 terms): x^2 + 3x, 2y - 4
  • Trinomials (3 terms): x^3 + 2x^2 - 3x, 2y^3 - 4y^2 + 5y

Operations

• Polynomials can be added, subtracted, and multiplied using the standard rules of algebra.
• Example:
  • (2x^2 + 3x) + (x^2 - 2x) = 3x^2 + x
  • (2x^2 + 3x) - (x^2 - 2x) = x^2 + 5x
  • (2x^2 + 3x) * (x^2 - 2x) = 2x^4 - 7x^3 + 6x^2

Simplification

• Polynomials can be simplified by combining like terms.
• Example:
  • 2x^2 + 3x + x^2 = 3x^2 + 3x
  • x^2 - 2x + 3x = x^2 + x

Definition of Polynomials

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, with variables raised to non-negative integer powers.

Terms of Polynomials

• A term is a single element of a polynomial, consisting of a coefficient and a variable(s) raised to a power.
• Examples of terms: 3x, 2x^2, 5

Degrees of Polynomials

• The degree of a term is the sum of the powers of the variables.
• The degree of a polynomial is the highest degree of its terms.
• Examples of degrees:
  • The degree of 3x is 1.
  • The degree of 2x^2 is 2.
  • The degree of 4x^2 + 2x + 1 is 2.

Classification of Polynomials

• Polynomials can be classified based on their degree:
  • Monomials (degree 1): 2x, 3y
  • Binomials (degree 2): x^2 + 3x, 2y^2 - 4y
  • Trinomials (degree 3): x^3 + 2x^2 - 3x, 2y^3 - 4y^2 + 5y
• Polynomials can also be classified based on the number of terms:
  • Monomials (1 term): 2x, 3
  • Binomials (2 terms): x^2 + 3x, 2y - 4
  • Trinomials (3 terms): x^3 + 2x^2 - 3x, 2y^3 - 4y^2 + 5y

Operations with Polynomials

• Polynomials can be added, subtracted, and multiplied using the standard rules of algebra.
• Examples of operations:
  • (2x^2 + 3x) + (x^2 - 2x) = 3x^2 + x
  • (2x^2 + 3x) - (x^2 - 2x) = x^2 + 5x
  • (2x^2 + 3x) * (x^2 - 2x) = 2x^4 - 7x^3 + 6x^2

Simplification of Polynomials

• Polynomials can be simplified by combining like terms.
• Examples of simplification:
  • 2x^2 + 3x + x^2 = 3x^2 + 3x
  • x^2 - 2x + 3x = x^2 + x