Honors Pre-Calculus Unit 3 Review: Polynomial and Rational Functions

Key Concepts

• Polynomials consist of terms involving variables raised to non-negative integer powers and coefficients
• Degree of a polynomial determined by the highest power of the variable in the polynomial
• Leading coefficient is the coefficient of the term with the highest degree
• Standard form of a polynomial has the terms arranged in descending order of degree
• Polynomial functions can be represented graphically, with the degree determining the shape and behavior of the graph
• Rational functions formed by dividing one polynomial by another, resulting in a function with a non-polynomial denominator
  • Expressed as $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$
• Zeros of a polynomial are the x-intercepts of its graph and the solutions to the equation $P(x) = 0$

Types of Polynomials

• Monomial is a polynomial with only one term (e.g., $3x^2$, $-7x$, $5$)
• Binomial is a polynomial with exactly two terms (e.g., $x^2 + 3x$, $2x - 1$)
• Trinomial is a polynomial with exactly three terms (e.g., $x^2 + 2x + 1$, $3x^3 - 2x + 5$)
• Quadratic polynomial has a degree of 2 and can be written in the form $ax^2 + bx + c$, where $a \neq 0$
• Cubic polynomial has a degree of 3 and can be written in the form $ax^3 + bx^2 + cx + d$, where $a \neq 0$
• Quartic polynomial has a degree of 4 and can be written in the form $ax^4 + bx^3 + cx^2 + dx + e$, where $a \neq 0$
• Higher degree polynomials include quintic (degree 5), sextic (degree 6), septic (degree 7), and so on

Graphing Polynomials

• The degree of a polynomial determines the shape and behavior of its graph
  • Even degree polynomials have graphs that extend either up or down on both ends
  • Odd degree polynomials have graphs that extend in opposite directions on each end
• The leading coefficient affects the orientation of the graph
  • Positive leading coefficient results in a graph that opens upward for even degree polynomials and rises to the right for odd degree polynomials
  • Negative leading coefficient results in a graph that opens downward for even degree polynomials and falls to the right for odd degree polynomials
• Multiplicity of a zero determines the behavior of the graph at the x-intercept
  • Odd multiplicity results in the graph crossing the x-axis at the zero
  • Even multiplicity results in the graph touching the x-axis at the zero without crossing
• Graphing techniques include factoring, finding zeros, and identifying the end behavior based on the degree and leading coefficient
• Transformations of polynomial functions involve shifts, reflections, stretches, and compressions of the base graph

Roots and Zeros

• Roots of a polynomial are the values of x that make the polynomial equal to zero
• Zeros of a polynomial are the x-intercepts of its graph and correspond to the roots of the polynomial
• Multiplicity of a root or zero indicates how many times the factor $(x - r)$ appears in the factored form of the polynomial, where $r$ is the root or zero
  • Simple root or zero has a multiplicity of 1
  • Double root or zero has a multiplicity of 2
  • Triple root or zero has a multiplicity of 3, and so on
• Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots, counting multiplicities
• Rational Root Theorem helps find potential rational roots of a polynomial by considering the factors of the constant term and the leading coefficient
• Descartes' Rule of Signs provides an upper bound for the number of positive and negative real roots of a polynomial based on the sign changes in the coefficients

Rational Functions

• Rational functions are formed by dividing one polynomial by another, resulting in a function with a non-polynomial denominator
• Domain of a rational function consists of all real numbers except those that make the denominator equal to zero
• Vertical asymptotes occur at the x-values that make the denominator of a rational function equal to zero
• Horizontal asymptote exists when the degree of the numerator is less than the degree of the denominator
  • The horizontal asymptote is the line $y = 0$
• Oblique (slant) asymptote exists when the degree of the numerator is one less than the degree of the denominator
  • The oblique asymptote is a linear function determined by long division of the numerator by the denominator
• Graphing rational functions involves finding the domain, vertical and horizontal asymptotes, x- and y-intercepts, and analyzing the behavior of the function near the asymptotes and intercepts

Asymptotes and Holes

• Vertical asymptotes occur at the x-values that make the denominator of a rational function equal to zero
  • The graph approaches positive or negative infinity as x approaches the vertical asymptote from either side
• Horizontal asymptote exists when the degree of the numerator is less than the degree of the denominator
  • The graph approaches the horizontal asymptote as x approaches positive or negative infinity
• Oblique (slant) asymptote exists when the degree of the numerator is one less than the degree of the denominator
  • The graph approaches the oblique asymptote as x approaches positive or negative infinity
• Removable discontinuity (hole) occurs when a factor in the numerator and denominator cancels out, resulting in an undefined point on the graph
  • The hole is located at the x-value that makes both the numerator and denominator equal to zero
• Non-removable discontinuity occurs when a factor in the denominator does not cancel out with a factor in the numerator, resulting in a vertical asymptote
• Analyzing the behavior of a rational function near its asymptotes and holes helps in sketching an accurate graph

Applications in Real Life

• Polynomial functions model various real-world situations, such as the height of a projectile over time or the volume of a box with varying dimensions
• Quadratic functions describe the path of an object under the influence of gravity, such as a ball being thrown or a rocket launched
• Cubic functions model the volume of a rectangular box with a square base and varying height
• Higher degree polynomial functions approximate complex curves and surfaces in computer graphics and animation
• Rational functions model situations involving rates, such as the concentration of a drug in the bloodstream over time or the population growth of a species with limited resources
• Inverse proportionality relationships, such as the time it takes to complete a task with varying numbers of workers, can be modeled using rational functions
• Polynomial and rational functions are used in economics to analyze cost, revenue, and profit functions for businesses

Common Mistakes and How to Avoid Them

• Forgetting to consider the multiplicity of roots when factoring polynomials
  • Always check for repeated factors and express them with the appropriate exponent
• Misidentifying the degree of a polynomial by not considering the highest power of the variable
  • Carefully identify the term with the highest degree and use that to determine the overall degree of the polynomial
• Incorrectly applying the rules for transformations of polynomial functions
  • Remember that horizontal shifts are opposite the sign of the constant term inside the function, while vertical shifts follow the sign of the constant term outside the function
• Failing to find all the vertical asymptotes of a rational function by not factoring the denominator completely
  • Factor the denominator fully and set each factor equal to zero to find all the potential vertical asymptotes
• Misinterpreting the end behavior of a polynomial function based on its degree and leading coefficient
  • Use the rules for even and odd degree polynomials and the sign of the leading coefficient to correctly identify the end behavior
• Incorrectly canceling terms in a rational function, leading to incorrect simplification
  • Only cancel factors common to both the numerator and denominator, not individual terms
• Neglecting to consider the domain restrictions when graphing rational functions
  • Always identify the values of x that make the denominator equal to zero and exclude them from the domain