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College Algebra
Table of Contents
Unit 5 Overview: Polynomial and Rational Functions
5.1 Quadratic Functions
5.2 Power Functions and Polynomial Functions
5.3 Graphs of Polynomial Functions
5.4 Dividing Polynomials
5.5 Zeros of Polynomial Functions
5.6 Rational Functions
5.7 Inverses and Radical Functions
5.8 Modeling Using Variation

Polynomial functions are mathematical expressions that combine variables and coefficients using basic operations. These functions have unique graphical features like zeros, end behavior, and turning points. Understanding these elements helps predict a polynomial's shape and behavior.

Graphing polynomials involves identifying key characteristics and connecting them smoothly. Factors like degree, leading term, and zero multiplicity influence the graph's appearance. Techniques like factoring and applying the Intermediate Value Theorem aid in analyzing polynomial behavior and finding solutions.

Polynomial Function Graphs

Key features of polynomial graphs

  • Zeros (x-intercepts) where the graph crosses the x-axis and the function equals zero
  • End behavior describes the graph's trend as x approaches positive or negative infinity
    • Determined by the leading term's degree (even or odd) and coefficient (positive or negative)
    • Even degree graphs have both ends approach positive or negative infinity together (parabola, quartic)
    • Odd degree graphs have one end approach positive infinity and the other negative infinity (cubic, quintic)
  • Turning points are local maxima and minima where the graph changes direction
    • Occur at critical points where the derivative equals zero or is undefined (peaks, valleys)
  • Roots (also known as zeros) are the x-values where the polynomial function equals zero

Factoring for polynomial zeros

  • Factor the polynomial function to find its zeros by setting it equal to zero and factoring the expression
  • Zeros are the x-values that make the factored expressions equal zero
  • Factoring techniques include greatest common factor (GCF), grouping, trinomial factoring, difference of squares, sum and difference of cubes

Polynomial degree and graph characteristics

  • Degree is the highest power of the variable in the polynomial
  • Number of turning points is at most one less than the degree (Degree - 1)
  • Number of possible real zeros is at most equal to the degree, exactly equal if all zeros are real
  • Even degree graphs are symmetric about the y-axis (parabola)
  • Odd degree graphs are symmetric about the origin (cubic)

Graphing polynomials using properties

  1. Identify the leading term and determine the end behavior

  2. Find the y-intercept by evaluating the function at x = 0

  3. Calculate the zeros by factoring or using other methods

  4. Determine the multiplicity of each zero

    • Odd multiplicity means the graph crosses the x-axis (linear factor)
    • Even multiplicity means the graph touches the x-axis without crossing (quadratic factor)

  5. Locate turning points by finding critical points where the derivative equals zero or is undefined

  6. Connect key points using smooth curves, considering the end behavior and turning points

Zero multiplicity in polynomial graphs

  • Multiplicity is the number of times a zero is repeated in the factored form of the polynomial
  • Simple zero (multiplicity 1) means the graph crosses the x-axis at this point (linear factor)
  • Multiple zero (multiplicity greater than 1) means the graph touches the x-axis without crossing
    • Higher multiplicity results in a "flatter" touch point (quadratic factor for multiplicity 2, cubic factor for multiplicity 3)

Intermediate Value Theorem application

  • Intermediate Value Theorem (IVT) states if a polynomial function $f(x)$ is continuous on the closed interval $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists at least one value $c$ in $[a, b]$ such that $f(c) = k$
  • All polynomial functions are continuous on their entire domain
  • IVT helps determine the existence of zeros in a given interval and is useful for approximating zeros using methods like bisection or the Newton-Raphson method

Predicting polynomial behavior

  • Leading term determines the end behavior and general shape of the graph
  • Odd degree graphs have at least one real zero and opposite end behavior (cubic, quintic)
  • Even degree graphs may have no real zeros and similar end behavior (parabola, quartic)
  • Coefficients affect the graph's steepness, y-intercept, and horizontal shifts
  • Analyze the equation by identifying the degree, leading term, and coefficients
    • Factor the polynomial, if possible, to determine potential zeros
    • Consider the impact of each term on the graph's shape and behavior

Polynomial Functions and Their Properties

  • A function is a relation that assigns each input to exactly one output
  • A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication
  • The degree of a polynomial is the highest power of the variable in the expression
  • Coefficients are the numerical factors of the terms in a polynomial
  • Continuity is a key property of polynomial functions, ensuring smooth, unbroken graphs

Key Terms to Review (51)

Absolute value function: An absolute value function is a type of piecewise function that returns the non-negative value of its input. It is denoted as $f(x) = |x|$ and has a V-shaped graph.
Binomial coefficient: A binomial coefficient is a coefficient of any of the terms in the expansion of a binomial raised to a power, typically written as $\binom{n}{k}$ or $C(n,k)$. It represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order.
Degree: The degree of a polynomial is the highest power of the variable in its expression. It determines the most significant term when expanding or simplifying the polynomial.
Difference of squares: The difference of squares is a specific type of polynomial that takes the form $a^2 - b^2$, which can be factored into $(a + b)(a - b)$. It is based on the property that the product of a sum and a difference of two terms results in the difference of their squares.
Continuous: A function is continuous if there are no breaks, holes, or jumps in its graph. In other words, you can draw the entire graph without lifting your pen.
Cubic functions: A cubic function is a polynomial function of degree three, typically expressed in the form $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants and $a \neq 0$. These functions can have up to three real roots and exhibit distinct characteristics such as points of inflection.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Greatest common factor: The greatest common factor (GCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It is useful in simplifying fractions, factoring polynomials, and solving equations.
Local minimum: A local minimum is a point on the graph of a function where the function value is lower than all nearby points. It represents the lowest value within a specific interval.
Parabola: A parabola is a symmetric curve that represents the graph of a quadratic function. It can open upward or downward depending on the sign of the quadratic coefficient.
Multiplicity: Multiplicity of a root in a polynomial function is the number of times that root occurs. It affects the shape and behavior of the graph at the corresponding $x$-intercept.
Leading coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.
Local maximum: A local maximum of a function is a point at which the function's value is higher than that of any nearby points. It is not necessarily the highest point on the entire graph, but rather within a specific interval.
Polynomial: A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents. It can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
Roots: Roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as solutions or zeros of the equation.
Zeros: Zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, they are the solutions to the equation $P(x) = 0$.
Bisection Method: The bisection method is a numerical technique used to approximate the roots of a continuous function. It involves repeatedly dividing an interval in half and selecting the subinterval that contains the root, thereby narrowing down the search for the root.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the magnitude or strength of the relationship between the variable and the overall expression. Coefficients are essential in various mathematical contexts, including polynomial factorization, linear equations, quadratic equations, and the graphing of polynomial functions.
Cubic Function: A cubic function is a polynomial function of degree three, where the highest exponent of the variable is three. Cubic functions have a distinctive S-shaped curve and can exhibit a variety of behaviors, including having one, two, or three real zeros, depending on the coefficients of the function.
Continuity: Continuity is a fundamental concept in mathematics that describes the smooth and uninterrupted behavior of a function or graph. It is a crucial property that ensures a function's values change gradually without any abrupt jumps or breaks.
Critical Points: Critical points are the points on the graph of a function where the derivative of the function is equal to zero or undefined. These points represent local maxima, local minima, or points of inflection, which are crucial in understanding the behavior and properties of polynomial functions.
Degree: In mathematics, the term 'degree' refers to the measure of a polynomial or the measure of an angle. It is a fundamental concept that underpins various topics in algebra, trigonometry, and calculus, including polynomials, power functions, graphs, and trigonometric functions.
End Behavior: The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity. It describes the limiting values or patterns that the function exhibits as it extends towards the far left and right sides of its graph.
Derivative: The derivative is a fundamental concept in calculus that measures the rate of change of a function at a particular point. It represents the slope of the tangent line to the function's graph at that point, providing information about the function's behavior and how it is changing.
Difference of Squares: The difference of squares is a special type of polynomial expression where the terms are the difference between two perfect squares. This concept is particularly important in the context of factoring polynomials, working with rational expressions, solving quadratic equations, and understanding the properties of power functions and polynomial functions.
Function: A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions are a fundamental concept in mathematics and are essential in understanding various topics in college algebra, including coordinate systems, quadratic equations, polynomial functions, and modeling using variation.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides two or more integers without a remainder. It is a fundamental concept in algebra that is essential for understanding real numbers, factoring polynomials, working with rational expressions, solving quadratic equations, and analyzing power and polynomial functions.
Grouping: Grouping is the process of combining or organizing mathematical expressions, functions, or elements into a single unit to simplify operations, enhance readability, or perform specific calculations. It is a fundamental concept in mathematics that is particularly relevant in the contexts of rational expressions and the graphs of polynomial functions.
Intermediate Value Theorem: The Intermediate Value Theorem states that if a continuous function takes on two different values, it must also take on all intermediate values between those two values. In other words, if a function is continuous on an interval and takes on different values at the endpoints of that interval, then it must take on every value in between those endpoint values somewhere within the interval.
Leading Term: The leading term of a polynomial function is the term with the highest exponent of the variable. It is the most significant term in the function and often determines the overall behavior and properties of the polynomial.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It is the first, or leading, coefficient in the standard form of a polynomial expression. The leading coefficient plays a crucial role in understanding and analyzing various topics in college algebra, including polynomials, quadratic equations, functions, and more.
Multiple Zero: A multiple zero, also known as a repeated root or repeated zero, is a value of the independent variable for which a polynomial function evaluates to zero, and this value occurs more than once in the function's set of zeros. In other words, it is a point where the graph of the polynomial function touches or crosses the x-axis more than once.
Local Maxima: A local maximum is a point on the graph of a function where the function value is greater than or equal to the function values at all nearby points. It represents a peak or highest point in a specific region of the graph.
Local Minima: A local minimum is a point on a function's graph where the function value is lower than the function values at all nearby points. It represents a point where the function has a smallest value in its immediate vicinity, even if it may not be the absolute lowest value of the function across its entire domain.
Multiplicity: Multiplicity refers to the number of times a particular value, known as a zero or root, occurs in the factorization of a polynomial function. It is an important concept in understanding the behavior of polynomial graphs and the nature of their zeros.
Newton-Raphson Method: The Newton-Raphson method is an iterative algorithm used to find the roots or zeros of a function. It is a widely used technique in numerical analysis and is particularly effective for finding solutions to nonlinear equations. The method is based on the idea of using the slope of the function at a given point to estimate the location of the next approximation to the root.
Odd Degree: In the context of polynomial functions, the term 'odd degree' refers to a polynomial where the highest exponent or power of the variable is an odd integer. This characteristic has important implications for the shape and behavior of the graph of the polynomial function.
Parabola: A parabola is a curved, U-shaped line that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola, and has many important applications in mathematics, science, and engineering.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in various areas of mathematics, including algebra, calculus, and the study of functions.
Quintic Function: A quintic function is a polynomial function of degree five, meaning it has the form $f(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, where $a$, $b$, $c$, $d$, $e$, and $f$ are real numbers. Quintic functions are part of the broader category of polynomial functions, which are widely studied in the context of graphing polynomial functions.
Quartic Function: A quartic function is a polynomial function of degree four, meaning it has the general form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$, where $a$, $b$, $c$, $d$, and $e$ are real numbers. Quartic functions are a specific type of polynomial function that are important in the study of graphs of polynomial functions and the determination of their zeros.
Roots: In mathematics, the term 'roots' refers to the solutions or values of a polynomial equation that make the equation equal to zero. Roots are an essential concept in various topics related to polynomial functions and equations, including quadratic equations, power functions, and the graphs of polynomial functions.
Simple Zero: A simple zero of a polynomial function is a real number that makes the function equal to zero and has a multiplicity of one. In other words, it is a point where the graph of the polynomial function crosses the x-axis and the function changes sign at that point.
Trinomial Factoring: Trinomial factoring is the process of breaking down a polynomial expression with three terms into the product of two or more simpler expressions. This technique is crucial in understanding the behavior and properties of polynomial functions, particularly in the context of graphing and analyzing their characteristics.
Sum and Difference of Cubes: The sum and difference of cubes is a mathematical concept that describes the relationship between the cubes of two numbers. It provides a way to simplify and manipulate expressions involving the cubes of variables or numbers.
Turning Points: Turning points are the critical points on the graph of a polynomial function where the direction of the curve changes. These points mark the transition between increasing and decreasing behavior, and are essential in understanding the overall shape and behavior of polynomial functions.
X-Intercepts: The x-intercepts of a graph are the points where the graph crosses the x-axis, or where the function's y-value is equal to zero. They represent the values of x for which the function has a solution of y = 0, and they provide important information about the behavior and characteristics of the function.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.
Zeros: Zeros, also known as roots, are the values of the independent variable that make a function equal to zero. They are the points where the graph of a function intersects the x-axis, representing the solutions to the equation f(x) = 0.
Even degree: An even degree refers to the highest power of a polynomial function that is an even number, such as 0, 2, 4, etc. When graphed, polynomial functions with an even degree exhibit particular characteristics, including a symmetric shape about the y-axis and a tendency to rise in both directions as x approaches positive and negative infinity. Understanding this concept helps to predict the behavior and shape of polynomial graphs effectively.