Polynomial & Rational Function Behavior – Full Study Notes

What Makes an Expression a Polynomial

• MUST have only positive, whole-number exponents on every variable.
  • No negative exponents, no fractional exponents.
  • A variable may not appear in a denominator; that would make the expression irrational / not a polynomial.
• Vocabulary
  • Degree = largest exponent appearing anywhere in the expression.
  • Leading coefficient = coefficient that is attached to that largest exponent after the polynomial is written in standard form.

Classifying an Expression ("Yes/No" Test)

1. Scan every exponent.
   • All positive integers? → keep going.
   • If any exponent < 0, fractional, or if a variable sits in a denominator → NOT a polynomial.
2. If "Yes, it is a polynomial": state
   • Degree (largest exponent).
   • Leading coefficient (number attached to that largest exponent).

Example check-ups (from class):

• 8x^5 - 3x^2 + 4 → YES, degree 5, leading coeff 8.
• \tfrac12x^2 + 7x + 9 → YES, degree 2, leading coeff \tfrac12.

End Behavior of a Polynomial

Only TWO things control what the far-left and far-right tails do:

1. The degree (even vs. odd).
2. The sign of the leading coefficient (positive vs. negative).

Quick procedure for any polynomial:

1. Locate the term with largest exponent.
2. Note whether that exponent is even or odd.
3. Note the sign (+/–) on its coefficient.
4. Pick the correct verbal pair from the chart.

Example: -7x^4 + 2x^3 - 9
• Largest exponent 4 (even) — coefficient −7 (neg.) → Fall left, fall right.

Inside Behavior – Zeros & Multiplicity

Concept links

• Zero / x-intercept / root / solution are synonyms → set y = 0 and solve for x.
• Multiplicity = the power on a factor once the polynomial is fully factored.
  • Even multiplicity → the graph touches (bounces) and turns around at that x-value.
  • Odd multiplicity → the graph crosses the x-axis at that x-value.

Finding zeros when already factored:

1. Take each factor, set its inside equal to zero.
2. Solve for x.
3. The outside exponent on that factor = multiplicity.

Illustrative examples

• 9(x-2)(x+5)
  Zeros: x=2,\;x=-5 each with multiplicity 1 → each crosses.
• x(x-6)^2
  Zeros: x=0 (mult 1, crosses) ; x=6 (mult 2, touches/bounces).

Tip: A pure constant like 9 in front never gives a zero because it does not contain x.

Factoring Reminder (appears on every test!)

• Always pull out a GCF first.
• If repeated identical binomials appear, write them as a single factor raised to a power; it prevents you from missing the correct multiplicity.

Example
x(36 - 36x + 9x^2) \;\to\; x(6 - 3x)^2
→ zeros at x=0 (mult 1) and x=2 (mult 2).

Rational Functions – Holes & Vertical Asymptotes (VAs)

Same overall theme: factor first.

Step-by-step algorithm

1. Factor numerator & denominator completely.
2. Cancel any identical factors.
   • Each factor that cancels ⇒ a hole (removable discontinuity).
   Set that factor = 0 to get x-coordinate of the hole.
3. After canceling, whatever factors remain in the denominator ⇒ vertical asymptotes.
   Set each remaining factor = 0; write answers as x = \text{constant}.
4. Anything that is neither a hole nor a VA is part of the domain.

Key notes

• You can only cancel whole factors (multiplication). Addition/subtraction inside a denominator locks the terms together.
• Both holes & VAs are domain restrictions; holes are removable, VAs are non-removable.

Example 1
\frac{x(x-1)}{x(x-1)(x-1)} → an x cancels.

• Hole: factor x=0.
• Remaining denom (x-1)^2 ⇒ VA at x=1.

Example 2
\frac{x+4}{x^2-2x} = \frac{x+4}{x(x-2)} (already factored)

• No common factor with numerator ⇒ no hole.
• Denominator pieces x=0,\;x=2 ⇒ VAs at x=0, x=2.

Example 3
\frac{(x+3)(x-5)}{(x+3)(x^2+19)}

• Factor x+3 cancels ⇒ hole at x=-3.
• Remaining denom x^2+19.
  Solving x^2+19=0 ⇒ x=\pm\sqrt{-19}=\pm i\sqrt{19} (imaginary).
  Non-real ⇒ no real VA.

Horizontal Asymptotes (HAs) – Three Memorize-Me Rules

Let n = degree (largest exponent) of numerator, m = degree of denominator.

1. If n < m (top smaller than bottom) → \boxed{y = 0}.
2. If n = m → divide the leading coefficients:
   \displaystyle y = \frac{an}{bm}.
3. If n > m (top bigger than bottom) → \boxed{\text{None}} (graph may have an oblique/slant asymptote, not tested here).

Remember:

• Horizontal lines are written y = …, vertical lines x = ….

Quick checks

1. \dfrac{6x+1}{x^2-4} → degrees 1 vs 2 → top smaller → y=0.
2. \dfrac{14x^2+9}{7x^2-5x+1} → degrees equal (2 vs 2) → divide leading coefficients 14/7=2 → y=2.
3. \dfrac{x^3-1}{x-4} → 3 vs 1 → top bigger → no horizontal asymptote.

Miscellaneous Connections & Test Tips

• Increasing vs. Decreasing behavior ties back to the sign of the leading coefficient for odd-degree polynomials (positive → rises to the right, negative → falls).
• ACT/SAT often swap the words “zeros,” “roots,” “x-intercepts,” “solutions” interchangeably – they all mean plug y=0.
• Horizontal vs. vertical asymptote equations must include the independent variable: write “y=…” or “x=…,” never just the number.
• Multiple-choice items frequently list all four verbs (fall/rise left/right). Read every word before selecting — one inversion changes the answer.
• Much of this unit is memorization; flash-card charts for end-behavior and asymptote rules are strongly recommended.