📈College Algebra Unit 5 – Polynomial and Rational Functions

Key Concepts and Definitions

• Polynomial functions are functions that consist of terms with non-negative integer exponents and coefficients
• Degree of a polynomial is the highest exponent of the variable in the polynomial
  • Leading coefficient is the coefficient of the term with the highest degree
• Standard form of a polynomial is written in descending order of degrees with like terms combined
• Rational functions are functions that can be written as the ratio of two polynomial functions $\frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$
• Domain of a rational function includes all real numbers except those that make the denominator equal to zero
• Vertical asymptotes occur when the denominator of a rational function equals zero
• Horizontal asymptotes describe the behavior of the function as $x$ approaches positive or negative infinity

Polynomial Functions: Structure and Properties

• Polynomial functions can be classified by their degree (linear, quadratic, cubic, etc.)
• Polynomials are closed under addition, subtraction, and multiplication
  • Sum, difference, or product of two polynomials is always another polynomial
• Polynomial long division and synthetic division are techniques used to divide polynomials
• Remainder Theorem states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is equal to $P(a)$
• Factor Theorem states that $(x - a)$ is a factor of a polynomial $P(x)$ if and only if $P(a) = 0$
• Fundamental Theorem of Algebra states that a polynomial of degree $n$ has exactly $n$ complex roots (including repeated roots)
• Polynomials with real coefficients have complex roots that occur in conjugate pairs

Graphing Polynomial Functions

• Polynomial functions can be graphed by plotting points or using transformations of parent functions
• End behavior of a polynomial function is determined by the leading term
  • For even degree polynomials, the ends of the graph will either both point up or both point down
  • For odd degree polynomials, one end will point up and the other will point down
• Multiplicity of a zero determines the behavior of the graph near the x-intercept
  • If the multiplicity is odd, the graph will cross the x-axis at the zero
  • If the multiplicity is even, the graph will touch the x-axis at the zero but not cross it
• Turning points (local maxima and minima) can be found using the first derivative test
• Polynomial functions are continuous and differentiable on their entire domain

Zeros and Roots of Polynomials

• Zeros (or roots) of a polynomial are the x-values where the function equals zero
• Multiplicity of a zero is the number of times the zero appears as a factor in the polynomial
• Rational root theorem provides a list of potential rational zeros of a polynomial with integer coefficients
• Descartes' Rule of Signs helps determine the number of positive and negative real zeros of a polynomial
• Complex zeros of a polynomial with real coefficients occur in conjugate pairs
• Polynomial functions can be factored using various methods (grouping, sum/difference of cubes, etc.)
• Quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ can be used to find the zeros of a quadratic polynomial

Rational Functions: Definition and Properties

• Rational functions are the ratio of two polynomial functions $\frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$
• Domain of a rational function excludes values that make the denominator equal to zero
• Vertical asymptotes occur when the denominator equals zero
  • To find vertical asymptotes, set the denominator equal to zero and solve for x
• Horizontal asymptotes describe the behavior of the function as $x$ approaches positive or negative infinity
  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0
  • If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is y = the ratio of the leading coefficients
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote
• Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator

Graphing Rational Functions

• To graph a rational function, first find the domain, vertical asymptotes, and horizontal asymptotes
• Plot any x-intercepts (zeros of the numerator) and y-intercepts (evaluate the function at x = 0)
• Use the sign of the function (positive or negative) in each interval of the domain to determine the behavior of the graph
• Holes in the graph occur when a factor cancels out in both the numerator and denominator
  • To find holes, factor the numerator and denominator and cancel common factors, then set the remaining denominator equal to zero
• Oblique asymptotes can be found by long division of the numerator by the denominator
• Sketch the graph by combining the asymptotes, intercepts, holes, and sign information

Applications and Problem Solving

• Polynomial and rational functions can model various real-world situations (population growth, resource allocation, etc.)
• Identify the key features of the problem and determine the appropriate type of function to use
• Create an equation to represent the situation using the given information
• Solve the equation or interpret the graph to answer questions about the situation
• Verify that the solution makes sense in the context of the problem
• Example application: A rectangular field has a perimeter of 200 meters. Express the area of the field as a function of its width, and determine the dimensions that maximize the area.

Common Mistakes and Tips

• Remember to use parentheses when substituting negative values into functions to avoid sign errors
• Be careful when canceling factors in rational expressions, as this may create holes in the graph
• When using the quadratic formula, double-check the sign of the discriminant ($b^2 - 4ac$) to determine the nature of the roots
• Make sure to consider the end behavior and asymptotes when graphing polynomial and rational functions
• Check your solutions by substituting them back into the original equation or evaluating the function at the solved values
• Practice identifying the key features of graphs (intercepts, asymptotes, turning points, etc.) to develop a strong understanding of function behavior
• When problem-solving, always interpret your final answer in the context of the situation to ensure it makes sense