PC40S

Unit 5: Polynomial Functions

Overview

What is a Polynomial Function?

  • A polynomial in one variable: \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where coefficients \( a_i \) are real and exponents are non-negative integers.
  • Allowed: addition, subtraction, multiplication by real constants; exponents must be whole numbers.
  • Not polynomials: negative/fractional exponents \((x^{-1}, \sqrt{x})\), variables in denominators/exponents, absolute values, trig functions.
  • Example: \( f(x) = 5x^3 - 3x + 7 \) (degree 3).

Key Vocabulary

  • Term: \( a_i x^i \)
  • Coefficient: \( a_i \)
  • Leading term: highest-degree term (e.g., \( 5x^3 \))
  • Leading coefficient: coefficient of leading term (e.g., \(5\))
  • Degree: highest exponent with non-zero coefficient
  • Constant term: \( a_0 \)
  • Zero/root/x-intercept: \( x = r \) such that \( f(r) = 0 \)
  • Multiplicity: number of times \( (x - r) \) divides the polynomial

Degree & Leading Coefficient

  • Degree: max number of \(x\)-intercepts (counting multiplicity), max turning points \(≤\) degree \(− 1\).
  • Leading coefficient: controls vertical scale and end behaviour (arrow directions) with degree parity.

End Behaviour by Degree & Leading Coefficient

Even degree, +LC

Both ends up
Even degree, −LC

Both ends down
Odd degree, +LC

Left down, right up
Odd degree, −LC

Left up, right down
  • Even degree: both ends same direction.
  • Odd degree: ends opposite directions.
  • +LC: right end up; −LC: right end down.

Typical Shapes by Degree

Linear (1)
Quadratic (2)
Cubic (3)
Quartic (4)
Quintic (5)

Root Behaviour (Multiplicity)

Simple root (1)

Crosses x-axis
Double root (2)

Bounces
Triple root (3)

Flattens through
  • Odd multiplicity: crosses (higher odd = flatter crossing).
  • Even multiplicity: touches/bounces.

Graphing Polynomial Functions

  1. Find \(x\)-intercepts (zeros) and their multiplicities.
  2. Find \(y\)-intercept: evaluate \( f(0) \).
  3. Determine end behaviour (degree & leading coefficient).
  4. Plot intercepts with correct root behaviour (cross/bounce/flatten).
  5. Draw end arrows, connect smoothly (max turning points ≤ degree \(− 1\)).
  6. Optionally, plot extra points for accuracy.
Example: \( f(x) = (x-2)^2(x+1) \)
Zeros: \( x = 2 \) (multiplicity 2, bounce), \( x = -1 \) (multiplicity \(1\), cross)
y-intercept: \( f(0) = 4 \)
Degree: 3 (odd), LC: +1 → left down, right up

Polynomial Division

  • Long division: divide by any polynomial, get quotient & remainder.
  • Synthetic division: shortcut for divisors \( x-a \).
Example: Divide \( 2x^3 - 3x^2 + 4x - 5 \) by \( x-2 \).
Long division: Quotient \( 2x^2 + x + 6 \), remainder \(7\).
Synthetic division: Coefficients: 2, −3, 4, −5; use \( a = 2 \).
Result: Quotient \( 2x^2 + x + 6 \), remainder \(7\).

Remainder & Factor Theorems

  • Remainder Theorem: remainder when dividing by \( x-a \) is \( f(a) \).
  • Factor Theorem: \( x-a \) is a factor iff \( f(a)=0 \).
  • Multiplicity: if \( (x-a)^k \) divides \( f(x) \), then a is a root of multiplicity \(k\).
Example: \( g(x) = (x+3)^2(x-1) \)
\( g(-3) = 0 \) → \( x+3 \) is a factor (multiplicity \(2\), bounce)
\( g(1) = 0 \) → \( x-1 \) is a factor (multiplicity \(1\), cross)

Properties & Quick Checks

  • Domain: all real numbers.
  • Range: depends on degree & LC (even: global min/max; odd: unbounded).
  • Graphs are smooth, continuous (no breaks/corners).
  • Max \(x\)-intercepts: degree (counting multiplicity).
  • Max turning points: degree \(− 1\).
  • Leading term test: for large \(|x|\), \( f(x) \sim a_n x^n \).

Worked Example: All-in-One

\( f(x) = - (x-1)^2 (x+2) (x-3) \)
Degree: \(4\) (even), LC: \(−1\) → both ends down
Zeros: \( x = 1 \) (multiplicity 2, bounce), \( x = -2 \) (multiplicity \(1\), cross), \( x = 3 \) (multiplicity \(1\), cross)
y-intercept: \( f(0) = 6 \)
Sketch: plot \( (-2, 0) \), \( (1, 0) \) bounce, \( (3, 0) \) cross, \( (0, 6) \); draw end arrows down, connect smoothly (max \(3\) turning points)