Graphs & Roots of Polynomial Functions (DP IB Analysis & Approaches (AA): HL): Revision Note

Sketching polynomial graphs

What is a polynomial?

• A polynomial is a sum of terms of the form $a{x}^k$ where

  • $a$ is a real number

  • $k\ge 0$ is an integer

• A polynomial looks like $P\left(x\right)=a_n{x}^n+...+a_2{x}^2+a_{1}x+a_0$

• The degree of a polynomial is its highest power

  • e.g. the degree of $4x^3+5x^2-7$ is 3

  • e.g. the degree of $2x-7$ is 1

• The leading term of a polynomial is the term with the highest power

  • e.g. the leading term of $4x^3+5x^2-7$ is $4x^3$

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

• If a real polynomial $P\left(x\right)$ has degree n

  • it will have n zeros which can be written in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$

    • these zeros are not necessarily distinct

    • they can be repeated

• For example:

  • A quadratic will have 2 zeros

  • A cubic function will have 3 zeros

  • A quartic will have 4 zeros

• Every real polynomial of odd degree has at least one real zero

What do I need to know to sketch the graph of a polynomial function?

• Suppose $P\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+ \dots +a_{1}x+a_0$ 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 $y=a_0$

  • 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 $x=k$ is a root with multiplicity m then ${(x-k)}^m$ 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 ± ∞

How do I sketch the graph of a polynomial function?

• Plot the y-intercept

• Plot the roots

• Identify the multiplicity of each root

• Identify where the graph starts and ends

• 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

    • 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

Solving polynomial equations

What is the fundamental theorem of algebra?

• The fundamental theorem of algebra states that 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?

• Let $P\left(x\right)=0$ where $P\left(x\right)$ is a real polynomial of degree n

  • $P\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+ \dots +a_{1}x+a_0$

• In your exam

  • you may be given one zero

  • or you might have to find a zero $x=k$

    • You can do this by substituting values into $P\left(x\right)$ until it equals 0

• If you know $x=k$ is a real root

  • then $\left(x-k\right)$ is a factor

• If you know $x=a+b\mathrm{i}$ is a non-real root

  • then you know a quadratic factor $\left(x-\left(a+b\mathrm{i}\right)\right)\left(x-\left(a-b\mathrm{i}\right)\right)$

    • This can be written as $\left(\left(x-a\right)-b\mathrm{i}\right)\left(\left(x-a\right)+b\mathrm{i}\right)$

    • And it simplifies to $\left({\left(x-a\right)}^2+b^2\right)$

• You can use polynomial division to divide $P\left(x\right)$ by a factor to get another factor

  • e.g. dividing a cubic by a linear factor will give you a quadratic factor

  • You then may be able to factorise this new factor

• You can solve the polynomial equation once you know all of its factors