Intro
Definition of Polynomials
A polynomial is an algebraic expression composed of terms, where each term includes a variable raised to a non-negative integer power and a coefficient.
Polynomials can have multiple terms and are generally written in descending order of powers.
Example: $3x^2 − 4x + 7$ is a polynomial with three terms: $3x^2$, $−4x$, and $7$.
Polynomial Terminology
Degree: The degree of a polynomial is the highest power of the variable. For instance, in, the degree is 3.
Leading Coefficient: The coefficient of the term with the highest degree. In , the leading coefficient is 5.
Constant Term: A term with no variable, such as +8
Operations with Polynomials
Addition/Subtraction: Combine like terms (terms with the same variable and power). For example
Multiplication: Distribute each term in one polynomial by each term in the other. Use the FOIL method for binomials or the distributive property for larger polynomials. Example:
Division: Divide a polynomial by a monomial or another polynomial, often using long division or synthetic division.
Factoring Polynomials
Greatest Common Factor (GCF): Factor out the largest term common to all terms. Example:
Factoring Trinomials: For trinomials in ax^2 + bx + c, find two numbers that multiply to ac and add to b.
Special Factoring Patterns:
– Difference of Squares:
– Perfect Square Trinomials:
Polynomial Functions
A polynomial function is defined by a polynomial equation, such as $f(x) = 2x^3 − x^2 + 4x − 5$. Polynomial functions are continuous with no breaks, holes, or gaps.
Roots/Zeros: The roots (or zeros) of a polynomial function are values of $x$ that make $f(x) = 0$. These can be found by factoring the polynomial and setting each factor to zero.