Algebra: 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.

Examples

• Simple polynomials: 2x, 3x^2, 5
• Polynomials with multiple terms: 2x + 3, x^2 - 4x + 2

Properties

• Degree: The highest power of the variable(s) in the polynomial. (e.g., 2x^2 + 3x has a degree of 2)
• Leading term: The term with the highest degree. (e.g., 2x^2 in 2x^2 + 3x)
• Leading coefficient: The coefficient of the leading term. (e.g., 2 in 2x^2)

Operations

• Addition: Polynomials can be added by combining like terms.
• Subtraction: Polynomials can be subtracted by combining like terms with opposite signs.
• Multiplication: Polynomials can be multiplied using the distributive property.

Special Types of Polynomials

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

Factoring Polynomials

• Factoring out the greatest common factor (GCF): Finding the GCF of all terms and dividing it out.
• Factoring quadratic expressions: Using the form x^2 + bx + c = (x + d)(x + e) to factor quadratic expressions.

Definition of Polynomials

• A polynomial consists of variables and coefficients combined using only addition, subtraction, and multiplication.
• Variables in a polynomial are raised to non-negative integer powers.

Characteristics of Polynomials

• Simple polynomials can be represented by a single term, such as 2x or 3x^2.
• Polynomials can have multiple terms, such as 2x + 3 or x^2 - 4x + 2.

Properties of Polynomials

• The degree of a polynomial is the highest power of the variable(s) in the polynomial.
• The leading term is the term with the highest degree in the polynomial.
• The leading coefficient is the coefficient of the leading term.

Polynomial Operations

• Polynomials can be added by combining like terms.
• Polynomials can be subtracted by combining like terms with opposite signs.
• Polynomials can be multiplied using the distributive property.

Special Types of Polynomials

• A monomial is a polynomial with only one term, such as 2x.
• A binomial is a polynomial with two terms, such as x + 2.
• A trinomial is a polynomial with three terms, such as x^2 + 2x + 1.

Factoring Polynomials

• Factoring out the greatest common factor (GCF) involves finding the GCF of all terms and dividing it out.
• Quadratic expressions can be factored using the form x^2 + bx + c = (x + d)(x + e).