Polynomials: Terms, Degree, and Classification

• A polynomial is an algebraic expression using variables, coefficients, and exponents.
• Polynomials use addition, subtraction, and multiplication.
• Exponents in polynomials must be whole numbers.
• Polynomials can't have variables in the denominator.

Terms, Degree, and Leading Coefficient

• A term is a monomial within a polynomial.
• In the polynomial , the terms are , , and .
• The degree of a term is the sum of its variables' exponents.
• has a degree of 4.
• The degree of a polynomial is the highest degree of any term in the polynomial.
• In , the degree is 4.
• The leading coefficient is the coefficient of the term with the highest degree.

Classification of Polynomials

• Polynomials are classified by the number of terms they contain:
• A monomial has one term.
• is a monomial.
• A binomial has two terms.
• is a binomial.
• A trinomial has three terms.
• Any expression with one or more terms is considered a polynomial.

Operations on Polynomials

• Like terms are combined in addition and subtraction.
• The expression simplifies to .

Multiplication

• Multiply each term of the polynomial by the monomial when multiplying a monomial by a polynomial.
• Use the distributive property (FOIL) when multiplying binomials.
• expands to .
• Distribute each term in the first polynomial to each term in the second when multiplying polynomials.

Special Products

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

Polynomial Division

• Long division is similar to numerical long division.
• Divide the first term, multiply, subtract, bring down the next term, and repeat these steps.
• Synthetic division simplifies dividing by a linear binomial in the form of .
• Write coefficients, apply division steps, and find the quotient and remainder.

Factoring Polynomials

• Factoring out the GCF involves identifying and extracting the highest common factor of all terms.
• 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. Understand terms, degree, and leading coefficients. Explore classifications: monomials, binomials, trinomials, and general polynomials.