Polynomials in Algebra

Polynomial

Definition

• A polynomial is an expression consisting of variables (such as x or y) and coefficients (numbers) combined using only addition, subtraction, and multiplication.
• The variables are raised to non-negative integer powers.

Types of Polynomials

• Monomial: A polynomial with only one term (e.g., 3x^2 or 5y)
• Binomial: A polynomial with two terms (e.g., x^2 + 3x or 2y^2 - 4y)
• Trinomial: A polynomial with three terms (e.g., x^2 + 2x + 1 or y^3 - 2y^2 + y)

Properties of Polynomials

• Degree: The highest power of the variable(s) in a polynomial (e.g., x^2 + 3x has a degree of 2)
• Leading Coefficient: The coefficient of the term with the highest degree (e.g., in x^2 + 3x, the leading coefficient is 1)
• Constant Term: The term with no variables (e.g., in x^2 + 3x + 2, the constant term is 2)

Operations with Polynomials

• Addition and Subtraction: Combine like terms (e.g., (x^2 + 2x) + (x^2 - 3x) = 2x^2 - x)
• Multiplication: Distribute each term in one polynomial to each term in the other polynomial (e.g., (x + 2) × (x + 3) = x^2 + 5x + 6)

Factoring Polynomials

• Factoring out the Greatest Common Factor (GCF): Divide each term by the GCF (e.g., 2x^2 + 4x = 2x(x + 2))
• Factoring Quadratic Expressions: Use the formula x^2 + bx + c = (x + d)(x + e) to factor quadratic expressions

Solving Polynomial Equations

• Linear Equations: Solve for the variable by isolating it on one side of the equation (e.g., 2x + 3 = 5 → x = 1)
• Quadratic Equations: Use factoring or the quadratic formula (x = (-b ± √(b^2 - 4ac)) / 2a) to solve quadratic equations

Polynomial

Definition

• An expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
• Variables are raised to non-negative integer powers.

Types of Polynomials

• Monomial: One-term expression (e.g., 3x^2 or 5y).
• Binomial: Two-term expression (e.g., x^2 + 3x or 2y^2 - 4y).
• Trinomial: Three-term expression (e.g., x^2 + 2x + 1 or y^3 - 2y^2 + y).

Properties of Polynomials

• Degree: Highest power of variables in a polynomial (e.g., x^2 + 3x has a degree of 2).
• Leading Coefficient: Coefficient of the term with the highest degree (e.g., in x^2 + 3x, the leading coefficient is 1).
• Constant Term: Term with no variables (e.g., in x^2 + 3x + 2, the constant term is 2).

Operations with Polynomials

• Addition and Subtraction: Combine like terms (e.g., (x^2 + 2x) + (x^2 - 3x) = 2x^2 - x).
• Multiplication: Distribute each term to each term in the other polynomial (e.g., (x + 2) × (x + 3) = x^2 + 5x + 6).

Factoring Polynomials

• Factoring out the GCF: Divide each term by the Greatest Common Factor (e.g., 2x^2 + 4x = 2x(x + 2)).
• Factoring Quadratic Expressions: Use the formula x^2 + bx + c = (x + d)(x + e) to factor quadratic expressions.

Solving Polynomial Equations

• Linear Equations: Isolate the variable on one side of the equation (e.g., 2x + 3 = 5 → x = 1).
• Quadratic Equations: Use factoring or the quadratic formula (x = (-b ± √(b^2 - 4ac)) / 2a) to solve quadratic equations.