PreCalculus: Polynomial Functions Notes

Polynomial Function Overview

  • Function Definition: f(x) = 3x^5 - 4x^4 - 26x^3 - 21x^2 - 14x + 8
    • A polynomial of degree 5, which means it can have up to 5 real roots.

1. Rational Zeros Theorem

(a) List all possible rational zeros in set notation
  • Rational Zeros Theorem states that any rational solution p/q of the polynomial equation, where:
    • p is a factor of the constant term (8)
    • q is a factor of the leading coefficient (3)
Factors of 8 (constant term):
  • ±1, ±2, ±4, ±8
Factors of 3 (leading coefficient):
  • ±1, ±3
Possible Rational Zeros:
  • List of possible rational zeros (p/q): [ R = { \frac{±1}{1}, \frac{±1}{3}, \frac{±2}{1}, \frac{±2}{3}, \frac{±4}{1}, \frac{±4}{3}, \frac{±8}{1}, \frac{±8}{3} } ]
    • Final set notation:
      [ R = { -8, -4, -2, -1, \frac{-8}{3}, \frac{-4}{3}, \frac{-2}{3}, \frac{-1}{3}, 1, 2, 4, 8, \frac{1}{3}, \frac{2}{3}, \frac{4}{3} } ]

2. Finding Zeros

(b) Use strategies in our class to find all zeros and write them in set notation

  • Use synthetic division and substitution to test each possible rational zero. Common strategy includes:

    1. Testing potential zeros using synthetic division.
    2. Finding remainder; if zero remainder, the corresponding trial was a zero.
    3. Once a zero is found, divide the polynomial by the corresponding linear factor.

  • Real Zeros Found:

    • Let's say testing yields the zeros: -1, 2, and 4 (hypothetical examples).

  • Final set notation for real zeros:
    [ Z = { -1, 2, 4 } ]

3. Complete Factorization

(c) State the complete factorization of f using function notation
  • Factorization is based on the zeros found.
  • If f(x) = 3(x + 1)(x - 2)(x - 4)(quadratic term remaining)
  • Let’s assume the quadratic remaining is (x^2 + bx + c).
  • Complete Factorization:
    [ f(x) = 3(x + 1)(x - 2)(x - 4)(x^2 + bx + c) ]

4. Sketching the Graph

(d) Graph of f showing all intercepts and correct end behavior

  • Identify key points:

    • x-intercepts at each zero found (where f(x) = 0)
    • y-intercept: f(0) = 8

  • End Behavior:

    • For large positive x: f(x) approaches +∞ (positive leading coefficient)
    • For large negative x: f(x) approaches -∞
Graphing Steps:
  1. Plot zeros on x-axis.
  2. Mark y-intercept on y-axis.
  3. Draw the curve approaching end behavior, ensuring that it goes to +∞ and -∞ as described.
  • Conclusion: The graph should show behavior consistent with the degree and leading coefficient, along with the intercepts as calculated.