Unit 5: Polynomial Functions

Core Definitions

• Polynomial: \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), exponents are whole numbers, coefficients are real.
• Degree: highest exponent with non-zero coefficient.
• Leading term / coefficient: highest-degree term and its coefficient.
• Constant term: \( a_0 \).
• Zero/root: \( r \) such that \( f(r) = 0 \).
• Multiplicity: \( k \) if \( (x - r)^k \) is a factor.

End Behaviour (Leading-Term Test)

• For large \( |x| \), \( f(x) \approx a_n x^n \).

Typical Visuals by Degree

Zero Behaviour (Multiplicity Rules)

• Odd \(k\): crosses; higher odd \(k\) = flatter crossing.
• Even \(k\): bounces.

Intercepts & Quick Compute

• x-intercepts: solve \( f(x) = 0 \). If factored, set each factor to zero.
  Example: \( f(x) = (x-2)^2(x+1) \) → \( x = 2 \) (mult 2, bounce), \( x = -1 \) (mult 1, cross).
• y-intercept: \( f(0) \).
  Example: \( f(0) = (−2)^2 \cdot 1 = 4 \) → \((0, 4)\).

Graphing Checklist (Exam Ready)

1. Put \( f(x) \) in factored or easily evaluable form.
2. Find: degree \( n \), LC, end behaviour.
3. Find all real zeros and their multiplicities.
4. Compute y-intercept \( f(0) \).
5. Plot key points; mark bounce/cross at each root.
6. Sketch with correct end arrows and smooth curves (polynomials are continuous and smooth).
7. Turning points ≤ \( n-1 \). Add extra points if shape is unclear.

Quick Facts to Remember

• Domain: all real numbers.
• Number of real zeros ≤ degree.
• If \(LC > 0\) and n even → global min exists; if \(LC < 0\) and \(n\) even → global max exists.
• Odd degree: range is all real numbers.

Polynomial Division

• Long division: works for any divisor. Align terms, divide, multiply, subtract, repeat.
  Result: \( f(x) = \text{divisor} \cdot \text{quotient} + \text{remainder} \).
• Synthetic division: only for divisors \( x-a \). Use \(a\), write coefficients, bring down, multiply, add, repeat.
  Use for linear divisors.

Remainder & Factor Theorems

• Remainder Theorem: remainder of dividing \( f(x) \) by \( x-a \) equals \( f(a) \).
• Factor Theorem: \( x-a \) is a factor of \( f(x) \) iff \( f(a) = 0 \).
• Multiplicity: if \( f(a) = 0 \) and synthetic division by \( x-a \) leaves remainder \(0\) repeatedly \(k\) times, then \( (x-a)^k \mid f(x) \).

Mini Workflows (Speed Patterns)

• Testing root \( a \): compute \( f(a) \). If 0, factor \( x-a \) using synthetic division.
• Building factorization: repeat synthetic division to peel off \( x-a \) factors (tracks multiplicity).
• End behaviour: read degree parity and LC sign; draw arrows first, then fit curve through intercepts with correct bounce/cross.

Compact Example (All Steps)