Polynomials: Definition, Terms, and Examples

• A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables

Polynomial Basics

• Polynomials are commonly written in the form: a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n, a_{n-1}, ..., a_1, a_0 are coefficients and x is the variable
• The exponents on the variables must be non-negative integers
• A polynomial in one variable is called a univariate polynomial, a polynomial in more than one variable is called a multivariate polynomial

Terminology

• Term: a term in a polynomial is a product of a coefficient and a variable raised to a non-negative integer power
• Coefficient: the constant multiplier of the variables in a term
• Degree: the degree of a term in a polynomial is the exponent of the variable in that term; the degree of a constant term is 0
• Leading coefficient: the coefficient of the term with the highest degree
• Constant term: a term with no variable factor

Degree of a Polynomial

• Univariate polynomial: the highest degree of any term in the polynomial
• Multivariate polynomial: the highest sum of degrees of variables in any term
• The degree of the polynomial 5x^3 + 2x^2 + x + 7 is 3 The polynomial 3x^2y^3 + xy + 5 has degree 5 (2+3) because the degree of the first term is 5, the degree of the second term is 2, and 5 is the largest

Types of Polynomials

• Monomial: a polynomial with one term (e.g., 5x^2)
• Binomial: a polynomial with two terms (e.g., 3x + 2)
• Trinomial: a polynomial with three terms (e.g., x^2 + 4x + 1)

Polynomial Operations

• Addition: combine like terms (terms with the same variable and exponent)
• Subtraction: distribute the negative sign and combine like terms
• Multiplication: distribute each term of one polynomial to each term of the other polynomial
• Division: polynomial long division or synthetic division

Polynomial Evaluation

• To evaluate a polynomial, substitute a given value for the variable and simplify

Polynomial Functions

• A polynomial function is a function defined by a polynomial expression
• The graph of a polynomial function depends on its degree and leading coefficient

Roots of Polynomials

• A root (or zero) of a polynomial is a value of the variable that makes the polynomial equal to zero
• Finding the roots of a polynomial is equivalent to solving the equation polynomial = 0

Factor Theorem

• The factor theorem states that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0
• If 'a' is a root of the polynomial f(x), then (x-a) is a factor of f(x)
• Conversely, if (x-a) is a factor of f(x), then 'a' is a root of f(x)

Remainder Theorem

• When a polynomial f(x) is divided by (x - a), the remainder is f(a)
• This theorem provides a quick way to evaluate a polynomial at a specific value, by finding the remainder of the division

Fundamental Theorem of Algebra

• A non-constant single-variable polynomial with complex coefficients has at least one complex root
• A polynomial of degree n has exactly n complex roots, counted with multiplicity

Rational Root Theorem

• Helps find potential rational roots of a polynomial equation with integer coefficients
• If a polynomial a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 has a rational root p/q (in lowest terms), then p must be a factor of a_0 and q must be a factor of a_n
• This theorem gives a list of possible rational roots that can be tested using synthetic division or direct substitution

Factoring Polynomials

• Factoring is the process of expressing a polynomial as a product of simpler polynomials or factors
• Common techniques include:
  • Factoring out the greatest common factor (GCF)
  • Using special factoring patterns (e.g., difference of squares, perfect square trinomials)
  • Trial and error
  • Grouping
• Reduces the polynomial into simpler expressions, which helps in solving polynomial equations and simplifying algebraic expressions

Solving Polynomial Equations

• Setting the polynomial equal to zero and finding the roots
• Methods include:
  • Factoring
  • Using the quadratic formula (for quadratic equations)
  • Synthetic division and the rational root theorem
  • Numerical methods (for higher-degree polynomials)

Graphing Polynomial Functions

• The graph of a polynomial function is a smooth, continuous curve
• Key features to consider when graphing:
  • Degree and leading coefficient (determine end behavior)
  • Roots (x-intercepts)
  • Y-intercept
  • Turning points (local maxima and minima)
  • Symmetry

Polynomial Inequalities

• Comparing a polynomial expression to a constant or another polynomial expression using inequality symbols (<, >, ≤, ≥)
• To solve, find the critical values (roots) of the polynomial, create a sign chart, and test intervals
• The solution set consists of the intervals where the polynomial satisfies the inequality

Applications of Polynomials

• Polynomials are used in various fields:
  • Engineering (curve fitting, optimization)
  • Physics (modeling motion, projectile paths)
  • Computer graphics (representing curves and surfaces)
  • Economics (cost functions, revenue models)
  • Statistics (regression analysis)

Special Polynomials

• Quadratic Polynomial: degree 2, form ax^2 + bx + c
• Cubic Polynomial: degree 3, form ax^3 + bx^2 + cx + d
• Quartic Polynomial: degree 4, form ax^4 + bx^3 + cx^2 + dx + e
• These have specific methods and formulas for finding roots and analyzing behavior

Polynomial Long Division

• A method for dividing one polynomial by another polynomial of equal or lesser degree
• Similar to long division with numbers
• Involves dividing, multiplying, subtracting, and bringing down terms until the degree of the remainder is less than the degree of the divisor

Synthetic Division

• A shortcut method for dividing a polynomial by a linear factor of the form (x - a)
• More efficient than long division
• Used to find roots and factor polynomials quickly

Irreducible Polynomials

• A polynomial that cannot be factored into polynomials of lower degree over a given field (e.g., real numbers, complex numbers) Example: x^2 + 1 is irreducible over the real numbers but reducible over the complex numbers

Multivariate Polynomials

• Polynomials with more than one variable
• Concepts like degree, terms, and operations extend to multivariate polynomials, but with added complexity
• Applications in multivariable calculus, optimization, and computer-aided design

Explore the basics of polynomials, including their definition, terminology, and examples. Learn about terms, coefficients, and degrees. Understand leading coefficients and constant terms in polynomial expressions.