Mathematics: Polynomial and Rational Functions

Polynomial Functions

• Function notation:
  $f(x) = a<em>n x^n + a</em>{n-1} x^{n-1} + … + a<em>1 x + a</em>0$
• Degree (n):
  • If n is odd:
  • Leading coefficient a_n > 0 indicates the graph rises to the right
  • Leading coefficient a_n < 0 indicates the graph falls to the right
  • If n is even:
  • Leading coefficient a_n > 0 indicates the graph rises to the right
  • Leading coefficient a_n < 0 indicates the graph falls to the right

Intervals and Sign Analysis

• Consider logarithmic aspects and sign intervals of the functions:

  • Example intervals:

  • $(- ext{∞}, -3)$, $(-3, 3)$

  • Analyze behavior as x approaches critical points:

  • When evaluating points such as x approaching -3 or 3, check the sign of the function or factors (e.g. $(x-3)(x+3)$) to determine intervals where the function is positive or negative.

Analyzing Functions with Asymptotes

• Vertical Asymptotes: Locations where the function becomes undefined, for example:
  • At $x = -3$, $x = 3$
• Horizontal Asymptote: Determines the value the function approaches as $x$ approaches infinity, example:
  • At $y = 0$

Function Evaluation

• To find specific points of interest such as intercepts:
  • $f(0) = -9$ helps identify the value of the function at zero
  • Roots and intercepts are crucial; for example, setting $f(x) = 0$ gives roots

Rational Functions and Their Behavior

• Analyzing rational functions of the form $f(x) = \frac{N(x)}{D(x)}$ where N and D are polynomials. Considerations:
  • Finding Roots: Set numerator to zero
  • Finding Vertical Asymptotes: Set denominator to zero
• Example polynomial factoring:
  • $x^2 - 2x - 15 = (x - 5)(x + 3)$

Partial Fraction Decomposition

• Breaking down complex rational functions to simpler components:

  • For example, find constants A and B if
    $\frac{5x - 1}{(x - 5)(x + 3)} = \frac{A}{x - 5} + \frac{B}{x + 3}$

• Solve for A and B by substituting and forming a system of equations resulting from matching coefficients:

  1. $A + B = 5$
  2. $-5A + 3B = -1$

• Example solution yielding:

  • A = 2, B = 3