Polynomials Definition and Characteristics

Definition

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Characteristics

• A polynomial can have any number of terms.
• Each term is a monomial (a single variable or a product of variables) multiplied by a coefficient (a number).
• The variables are raised to non-negative integer powers.
• There are no variables in the denominator (i.e., no fractions).
• No variables are inside a root symbol (e.g., no square roots).

Types of Polynomials

• Monomial: A polynomial with only one term.
• Binomial: A polynomial with two terms.
• Trinomial: A polynomial with three terms.

Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable(s) in the polynomial.
• For example, the degree of x^2 + 3x - 4 is 2, since the highest power of x is 2.

Operations on Polynomials

• Addition: Combine like terms by adding their coefficients.
• Subtraction: Combine like terms by subtracting their coefficients.
• Multiplication: Multiply each term in one polynomial by each term in the other polynomial.

Factoring Polynomials

• Factoring out the greatest common factor (GCF): Remove the GCF from each term.
• Factoring by grouping: Factor out a common binomial from pairs of terms.
• Factoring quadratic expressions: Use the formula x^2 + bx + c = (x + d)(x + e), where d and e are constants.

Solving Polynomial Equations

• Linear polynomials: Solve by adding or subtracting the same value to both sides.
• Quadratic polynomials: Solve using the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a.
• Higher-degree polynomials: Use factoring, the rational root theorem, or numerical methods.

Definition and Characteristics of Polynomials

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• A polynomial can have any number of terms, each being a monomial (a single variable or a product of variables) multiplied by a coefficient (a number).
• The variables are raised to non-negative integer powers, and there are no variables in the denominator (i.e., no fractions).
• No variables are inside a root symbol (e.g., no square roots).

Types of Polynomials

• Monomial: A polynomial with only one term.
• Binomial: A polynomial with two terms.
• Trinomial: A polynomial with three terms.

Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable(s) in the polynomial.
• For example, the degree of x^2 + 3x - 4 is 2, since the highest power of x is 2.

Operations on Polynomials

• Addition: Combine like terms by adding their coefficients.
• Subtraction: Combine like terms by subtracting their coefficients.
• Multiplication: Multiply each term in one polynomial by each term in the other polynomial.

Factoring Polynomials

• Factoring out the greatest common factor (GCF): Remove the GCF from each term.
• Factoring by grouping: Factor out a common binomial from pairs of terms.
• Factoring quadratic expressions: Use the formula x^2 + bx + c = (x + d)(x + e), where d and e are constants.

Solving Polynomial Equations

• Linear polynomials: Solve by adding or subtracting the same value to both sides.
• Quadratic polynomials: Solve using the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a.
• Higher-degree polynomials: Use factoring, the rational root theorem, or numerical methods.

Learn about the definition and characteristics of polynomials, including the rules for combining variables and coefficients.