Solving Equations with Complex Roots (Edexcel A Level Further Maths: Core Pure)

What are complex roots? 

• Complex numbers provide solutions for quadratic equations which have no real roots

• Complex roots occur when solving a quadratic with a negative discriminant
  • This leads to square rooting a negative number

How do we solve a quadratic equation with complex roots?

• We solve an equation with complex roots in the same way we solve any other quadratic equations
  • If in the form we can rearrange to solve
  • If in the form we can complete the square or use the quadratic formula
• We use the property along with a manipulation of surds

• When the coefficients of the quadratic equation are real, complex roots occur in complex conjugate pairs
  • If is a root of a quadratic with real coefficients then  is also a root
• When the coefficients of the quadratic equation are non-real, the solutions will not be complex conjugates
  • To solve these use the quadratic formula

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

• We can find the equation of the form  if you are given a complex root in the form 
  • We know that the complex conjugate is another root, 
  • This means that and are factors of the quadratic equation
  • Therefore 
    • Writing this as will speed up expanding
  • Expanding and simplifying gives us a quadratic equation where and are real numbers

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:
• A cubic equation of the form  can have either:
• 3 real roots
• Or 1 real root and a complex conjugate pair

• • A quartic equation of the form  will have one of the following cases for roots:
    • 4 real roots
    • 2 real and 2 nonreal (a complex conjugate pair)
    • 4 nonreal (two complex conjugate pairs)

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

Number of roots of a cubic function

Number of roots of a quartic function

How do we solve a cubic equation with complex roots?

• Steps to solve a cubic equation with complex roots
  • If we are told that is a root, then we know  is also a root
  • This means that  and  are factors of the cubic equation
  • Multiply the above factors together gives us a quadratic factor of the form 
  • We need to find the third factor 
  • Multiply the factors and equate to our original equation to get

• From there either
  • Expand and compare coefficients to find 
  • Or use polynomial division to find the factor 
• 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