UNDERSTANDING POLYNOMIALS (Math Methods Units 1 & 2)

Polynomial Form

• General form: P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0

• Key features:

  
  

  • Number of terms = n + 1

  • Degree = highest power of x

  • Leading coefficient a_n determines end behaviour

  • Not unique when P(x) = 0

  
  

End Behaviour

• Determined by leading term a_nx^n

• Positive a_n: graph rises right

• Even degree: ends in same direction

• Odd degree: ends in opposite directions

• Examples:

  
  

  • Cubic: y = ax^3 + bx + c

  • Quartic: y = ax^4 + bx^2 + c

  
  

Polynomial Graph Gallery

• Roots = solutions to P(x) = 0

• Roots = x-intercepts

• Total number of roots = degree of polynomial

• Use graph shape and intercepts to identify polynomial form

Roots of Polynomials

• Find roots by:

  
  

  • Factorisation

  • Null Factor Law

  • Quadratic formula (if degree ≤ 2)

  
  

• Roots help sketch and analyse graphs