Polynomial Functions (DP IB Analysis & Approaches (AA)) : Revision Note

Sketching Polynomial Graphs

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

Solving Polynomial Equations

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