What are complex roots?

• A quadratic equation can either have two real roots (zeros), a repeated real root or no real roots
  • This depends on the location of the graph of the quadratic with respect to the?x-axis
• If a quadratic equation has no real roots we would previously have stated that it has?no real solutions
  • The quadratic equation will have a?negative discriminant
  • This means taking the square root of a negative number
• Complex numbers provide solutions for quadratic equations that have?no real roots

How do we solve a quadratic equation when it has complex roots?

• If a quadratic equation takes the form?ax²?+?bx?+?c?= 0 it can be solved by either using the quadratic formula or completing the square
• If a quadratic equation takes the form?ax²?+?b?= 0 it can be solved by rearranging
• The property i = √-1 is used
• If the coefficients of the quadratic are real then the complex roots will occur in complex conjugate pairs
  • If?z?=?m?+?ni?(n?≠ 0)?is a root of a quadratic with real coefficients then?z* =?m?-?ni?is also a root
• The?real part?of the solutions will have the same value as the?x?coordinate of the turning point on the graph of the quadratic

• When the coefficients of the quadratic equation are?non-real, the solutions will?not?be complex conjugates
  • To solve these you can use the quadratic formula

How do we factorise a quadratic equation if it has complex roots?

• If we are given a quadratic equation in the form?az²?+?bz?+?c?= 0, where?a, b,?and?c?∈ ?,?a?≠ 0 we can use its complex roots to write it in?factorised form
  • Use the quadratic formula to find the two roots,?z?^?=?p?+?qi and?z*^?=?p?-?qi
  • This means that?z?– (p?+?qi) and?z?– (p?–?qi) must both be factors of the quadratic equation
  • Therefore we can write?az²?+?bz?+?c?= (z?– (p?+?qi))(?z?– (p?-?qi))
  • This can be rearranged into the form (z?–?p?–?qi)(z?–?p?+?qi)

How do we find a quadratic equation when given a complex root?

• If we are given a complex root in the form?z^?=?p?+?qi we can find the quadratic equation in the form?az²?+?bz?+?c?= 0, where?a, b,?and?c?∈ ?,?a?≠ 0
  • We know that the second root must be?z*^?=?p?-?qi
  • This means that?z?– (p?+?qi) and?z?– (p?–?qi) must both be factors of the quadratic equation
  • Therefore we can write?az²?+?bz?+?c?= (z?– (p?+?qi))(?z?– (p?-?qi))
  • Rewriting this as ((z?–?p)?–?qi))((z?–?p) +?qi)) makes expanding easier
  • Expanding this gives the quadratic equation?z²?– 2pz?+?(p²?+?q²)
    • a?= 1
    • b?= -2p
    • c?=?p²?+?q²
• This demonstrates the important property (x – z)(x – z*) =?x²_?- 2Re(z)x?+ |z|²

How many roots should a polynomial have?

• We know that every?quadratic?equation has?two roots?(not necessarily distinct or real)
• This is a particular case of a more general rule:
  • Every polynomial equation, with real coefficients, of degree?n?has?n?roots
  • The?n?roots are not necessarily all?distinct?and therefore we need to count any?repeated?roots that may occur individually
• From the above rule we can state the following:
  • 3?real?roots
    • Or 1?real?root and a complex?conjugate?pair

• 
  • 4?real?roots
  • 2?real?and 2?nonreal?(a complex conjugate pair)
  • 4?nonreal?(two complex conjugate pairs)

• When a polynomial of any degree has one complex root it will always also have the complex conjugate as a root

How do we solve a cubic equation with complex roots?

• Steps to solve a cubic equation with complex roots
  • If we are told thatp?+ qi is a root, then we know?p?- qi is also a root
  • This means that?z?– (p?+?qi) and?z?– (p?–?qi) must both be factors of the cubic equation
  • Multiplying the above factors together gives us a quadratic factor of the form (Az^2?+?Bz?+ C)
  • We need to find the third factor (z?- α)
  • From there either
    • Expand?and?compare coefficients?to find
    • Or use?polynomial division?to find the factor (z?- α)
  • Finally, write your?three roots?clearly

How do we solve a polynomial of any degree with complex roots?

• When asked to find the roots of any polynomial when we are given one, we use almost the same method as for a cubic equation
  • State the initial root and its conjugate and write their factors as a quadratic factor (as above) we will have two unknown roots to find, write these as factors (z?- α) and (z?- β)
  • The unknown factors also form a quadratic factor (z?- α)(z?- β)
  • Then continue with the steps from above, either?comparing coefficients?or using?polynomial division
    • If using polynomial division, then solve the quadratic factor you get to find the roots α and β

How do we solve polynomial equations with unknown coefficients?

• Steps to find unknown variables in a given equation when given a root:
  • Substitute?the given root?p?+ qi into the equation f(z) = 0
  • Expand?and?group?together the?real?and?imaginary?parts (these expressions will contain our unknown values)
  • Solve?as simultaneous equations to find the unknowns
  • Substitute?the values into the?original?equation
  • From here continue using the previously described methods for finding other roots for the polynomial

How do we factorise a polynomial when given a complex root?

• If we are given a root of a polynomial of any degree in the form?z?=?p?+?qi
  • We know that the complex conjugate,?z*=?p?–?qi is another root
  • We can write (z?– (p?+?qi)) and (?z?– (p?-?qi)) as two linear factors
    • Or rearrange into one quadratic factor
  • This can be multiplied out with another factor to find further factors of the polynomial
• For higher order polynomials more than one root may be given
  • If the further given root is complex then its complex conjugate will also be a root
  • This will allow you to find further factors