2.7.3 Polynomial Functions

In exams you’ll commonly be asked to sketch the graphs of different polynomial functions with and without the use of your GDC.

What’s the relationship between a polynomial’s degree and its zeros?

• If a real polynomial P(x) has degree n, it will have n zeros which can be written in the form a + bi, where a, b ∈ ℝ
  • For example:
    • A quadratic will have 2 zeros
    • A cubic function will have 3 zeros
    • A quartic will have 4 zeros
  • Some of the zeros may be repeated
• Every real polynomial of odd degree has at least one real zero

How do I sketch the graph of a polynomial function without a GDC?

• Suppose  is a real polynomial with degree n
• To sketch the graph of a polynomial you need to know three things:
• The y-intercept
• Find this by substituting x = 0 to get y = a₀
• The roots
• You can find these by factorising or solving y = 0
• The shape
• This is determined by the degree (n) and the sign of the leading coefficient (a_n)

How does the multiplicity of a real root affect the graph of the polynomial?

• The multiplicity of a root is the number of times it is repeated when the polynomial is factorised
• If  is a root with multiplicity m then  is a factor of the polynomial

• The graph either crosses the x-axis or touches the x-axis at a root x = k where k is a real number
  • If x = k has multiplicity 1 then the graph crosses the x-axis at (k, 0)
  • If x = k has multiplicity 2 then the graph has a turning point at (k, 0) so touches the x-axis 
    • If x = k has odd multiplicity m ≥ 3 then the graph has a stationary point of inflection at (k, 0) so crosses the x-axis
    • If x = k has even multiplicity m ≥ 4 then the graph has a turning point at (k, 0) so touches the x-axis

How do I determine the shape of the graph of the polynomial?

• Consider what happens as x tends to ± ∞
  • If a_n is positive and n is even then the graph approaches from the top left and tends to the top right
    • 
  • If a_n is negative and n is even then the graph approaches from the bottom left and tends to the bottom right
    • 
  • If a_n is positive and n is odd then the graph approaches from the bottom left and tends to the top right
    • and 
  • If a_n is negative and n is odd then the graph approaches from the top left and tends to the bottom right
    • and 
• Once you know the shape, the real roots and the y-intercept then you simply connect the points using a smooth curve
• There will be at least one turning point in-between each pair of roots
  • If the degree is n then there is at most n – 1 stationary points (some will be turning points)
    • Every real polynomial of even degree has at least one turning point
    • Every real polynomial of odd degree bigger than 1 has at least one point of inflection
  • If it is a calculator paper then you can use your GDC to find the coordinates of the turning points
  • You won’t need to find their location without a GDC unless the question asks you to

What is “The Fundamental Theorem of Algebra”?

• Every real polynomial with degree n can be factorised into n complex linear factors
• Some of which may be repeated
• This means the polynomial will have n zeros (some may be repeats)
• Every real polynomial can be expressed as a product of real linear factors and real irreducible quadratic factors
• An irreducible quadratic is where it does not have real roots
• The discriminant will be negative: b² – 4ac < 0
• If a + bi (b ≠ 0) is a zero of a real polynomial then its complex conjugate a – bi is also a zero
• Every real polynomial of odd degree will have at least one real zero

How do I solve polynomial equations?

• Suppose you have an equation P(x) = 0 where P(x) is a real polynomial of degree n
  • 
• You may be given one zero or you might have to find a zero x = k by substituting values into P(x) until it equals 0
• If you know a root then you know a factor
  • If you know x = k is a root then (x – k) is a factor
  • If you know x = a + bi is a root then you know a quadratic factor (x – (a + bi))( x – (a – bi))
    • Which can be written as ((x – a) - bi)((x – a) + bi) and expanded quickly using difference of two squares
• You can then divide P(x) by this factor to get another factor
  • For example: dividing a cubic by a linear factor will give you a quadratic factor
• You then may be able to factorise this new factor