Polynomials: Terms, Degree, and Operations

• A polynomial is an algebraic expression with variables, coefficients, and exponents, combined by addition, subtraction, and multiplication
• Exponents in polynomials must be whole numbers
• Polynomials cannot have variables in the denominator

Terms, Degree, and Leading Coefficient

• Term: A monomial within a polynomial
• Degree of a Term: The sum of the exponents of the variables in the term
• Degree of a Polynomial: The highest degree of any term in the polynomial
• Leading Coefficient: The coefficient of the term with the highest degree

Classification of Polynomials

• Monomial: A polynomial with one term
• Binomial: A polynomial with two terms
• Trinomial: A polynomial with three terms
• Polynomial: An expression with one or more terms

Operations on Polynomials

Addition and Subtraction

• Combine like terms (terms sharing the same variable and exponent)

Multiplication

• Monomial × Polynomial: Multiply each term of the polynomial by the monomial
• Binomial × Binomial: Use the distributive property (FOIL method)
• Polynomial × Polynomial: Distribute each term in the first polynomial to each term in the second

Special Products

• Square of a Binomial:
• Difference of Squares:
• Cube of a Binomial:

Polynomial Division

Long Division

• Similar to numerical long division; divide the first term, multiply, subtract, bring down the next term, and repeat

Synthetic Division

• A shortcut for dividing by a linear binomial of the form
• Write coefficients, apply division steps, and find the quotient and remainder

Factoring Polynomials

• Factoring out the GCF (Greatest Common Factor): Find the highest common factor of all terms and factor it out
• Factoring Trinomials: Find two numbers that multiply to the constant and add to the middle term's coefficient
• Factoring by Grouping: Group terms, factor out common binomials, and rewrite as a product
• Factoring Special Products: Recognize and apply special factoring formulas.

The Remainder and Factor Theorems

• Remainder Theorem: When a polynomial is divided by , the remainder is
• Factor Theorem: If , then is a factor of

Polynomial Functions and Their Graphs

• End Behavior: Determined by the leading term’s degree and coefficient
• Zeros (Roots): Values where , found by factoring or using the quadratic formula
• Multiplicity of Zeros: Determines how a graph touches or crosses the x-axis at each root
• Turning Points: A polynomial of degree can have up to turning points

The Binomial Theorem

• Expands using coefficients from Pascal’s Triangle:
• represents binomial coefficients.

Learn about polynomials, algebraic expressions with variables, coefficients, and exponents. This covers terms, degree, leading coefficients, and polynomial classification (monomial, binomial, trinomial). Also learn about operations like addition, subtraction, and multiplication.