📈AP Pre-Calculus Unit 1 – Polynomial and Rational Functions

Key Concepts

• Understand polynomial functions expressed in standard form as $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$
• Identify the degree of a polynomial function as the highest power of the variable in the function
• Recognize the leading coefficient as the coefficient of the highest degree term
• Determine the end behavior of a polynomial function based on the degree and leading coefficient
  • If the degree is even and the leading coefficient is positive, both ends of the graph will point up
  • If the degree is even and the leading coefficient is negative, both ends of the graph will point down
• Analyze the number of turning points (maximums and minimums) in relation to the degree of the polynomial
• Comprehend the Fundamental Theorem of Algebra, which states that a polynomial of degree $n$ has exactly $n$ complex roots
• Distinguish between real and complex roots of a polynomial function
• Apply polynomial long division and synthetic division to divide polynomials and find roots

Polynomial Functions Basics

• Define a polynomial function as a function consisting of terms with non-negative integer exponents
• Classify polynomials by their degree (linear, quadratic, cubic, quartic, etc.)
• Identify the coefficients and constant term in a polynomial function
• Understand the standard form of a polynomial function: $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$
  • $a_n$ represents the leading coefficient
  • $n$ represents the degree of the polynomial
• Perform addition, subtraction, and multiplication of polynomial functions
• Apply the FOIL method to multiply binomials (First, Outer, Inner, Last)
• Factor polynomials using various techniques (greatest common factor, grouping, trinomial factoring, difference of squares, sum and difference of cubes)
• Solve polynomial equations by factoring and setting each factor equal to zero

Graphing Polynomials

• Identify key features of polynomial graphs, including y-intercept, x-intercepts (roots or zeros), turning points (maximums and minimums), and end behavior
• Determine the y-intercept by evaluating the function at $x = 0$
• Find the x-intercepts by factoring the polynomial and setting each factor equal to zero
• Analyze the end behavior of a polynomial function based on the degree and leading coefficient
  • For odd degree polynomials, the ends of the graph will point in opposite directions
  • For even degree polynomials with a positive leading coefficient, both ends will point up
  • For even degree polynomials with a negative leading coefficient, both ends will point down
• Recognize that the maximum number of turning points is one less than the degree of the polynomial
• Sketch the graph of a polynomial function using the key features and end behavior

Roots and Zeros

• Define a root (or zero) of a polynomial function as a value of $x$ that makes the function equal to zero
• Apply the Fundamental Theorem of Algebra to determine the number of roots based on the degree of the polynomial
• Distinguish between real and complex roots
  • Real roots are roots that are real numbers
  • Complex roots are roots that are complex numbers (involving the imaginary unit $i$)
• Find the roots of a polynomial by factoring and setting each factor equal to zero
• Use the Rational Root Theorem to list potential rational roots of a polynomial
• Apply Descartes' Rule of Signs to determine the possible number of positive and negative real roots
• Understand the relationship between roots and x-intercepts of a polynomial graph
• Identify the multiplicity of a root based on its appearance in the factored form of the polynomial

Rational Functions

• Define a rational function as a function that can be written as the ratio of two polynomial functions: $\frac{P(x)}{Q(x)}$
• Identify the domain of a rational function as all real numbers except those that make the denominator equal to zero
• Simplify rational expressions by factoring and canceling common factors
• Perform addition, subtraction, multiplication, and division of rational functions
• Solve rational equations by finding a common denominator and setting the numerator equal to zero
• Graph rational functions by identifying key features such as vertical and horizontal asymptotes, x-intercepts, y-intercept, and holes
• Determine the end behavior of a rational function based on the degrees of the numerator and denominator
  • 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 $y = \frac{a_n}{b_n}$, where $a_n$ and $b_n$ are the leading coefficients of the numerator and denominator, respectively
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, and the function will have oblique (slant) asymptotes

Asymptotes and Holes

• Identify vertical asymptotes of a rational function by finding the values of $x$ that make the denominator equal to zero
• Determine horizontal asymptotes of a rational function based on the degrees of the numerator and denominator
  • 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 $y = \frac{a_n}{b_n}$, where $a_n$ and $b_n$ are the leading coefficients of the numerator and denominator, respectively
• Recognize oblique (slant) asymptotes when the degree of the numerator is one more than the degree of the denominator
• Identify holes (removable discontinuities) in the graph of a rational function by finding values of $x$ that make both the numerator and denominator equal to zero after factoring
• Understand that holes occur when a factor cancels out in the simplified form of the rational function
• Determine the coordinates of a hole by evaluating the simplified rational function at the x-value that causes the hole

Applications and Problem Solving

• Model real-world situations using polynomial and rational functions (population growth, projectile motion, optimization problems)
• Interpret the meaning of the degree, coefficients, and constants in the context of the problem
• Solve applied problems involving polynomial and rational functions
  • Determine the maximum or minimum value of a function in a given context
  • Find the zeros or roots of a function and interpret their meaning
• Analyze the behavior of a function based on its graph or equation in the context of the problem
• Use polynomial and rational functions to make predictions and decisions in real-world scenarios
• Combine polynomial and rational functions with other function types (exponential, logarithmic, trigonometric) to model more complex situations

Common Mistakes and Tips

• Be careful when simplifying rational expressions, ensuring that you factor both the numerator and denominator completely before canceling common factors
• Remember that when solving polynomial or rational equations, you must set each factor equal to zero to find all the solutions
• When graphing polynomial functions, make sure to consider the end behavior based on the degree and leading coefficient
• Don't forget to identify the multiplicity of roots when factoring polynomials, as this affects the behavior of the graph near the x-intercepts
• When graphing rational functions, pay attention to the vertical and horizontal asymptotes, as well as any holes in the graph
• Be mindful of the domain restrictions when working with rational functions, as division by zero is undefined
• When solving applied problems, always interpret the results in the context of the problem to ensure they make sense
• Double-check your work, especially when performing arithmetic operations on polynomials and rational expressions, to avoid simple mistakes