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?∈ ?
  • 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?

• 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 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?

• 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?

• 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?=?kis a rootthen?(x?–?k) is a factor
  • If you know?x?=?a?+?biis a rootthen 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