Complex Roots of Polynomials (CIE A Level Maths: Pure 3)

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Complex Roots of Quadratics

What are complex roots? 

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

8-1-4-complex-roots-of-polynomials-diagram-1

  • 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 a z squared plus b equals 0 blank open parentheses a not equal to 0 close parentheses we can rearrange to solve
    • If in the form a z squared plus b z plus c equals 0 blank left parenthesis a not equal to 0 right parenthesis we can complete the square or use the quadratic formula
  • We use the property straight i equals square root of negative 1 end root along with a manipulation of surds
    • square root of negative a end root equals square root of a cross times negative 1 end root equals square root of a cross times square root of negative 1 end root
  • When the coefficients of the quadratic equation are real, complex roots occur in complex conjugate pairs
    • If z equals m plus n straight i space left parenthesis straight n space not equal to space 0 right parenthesis is a root of a quadratic with real coefficients then z to the power of asterisk times equals m minus n straight i 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 bold italic z squared plus bold italic b bold italic z plus bold italic c equals 0 if you are given a complex root in the form bold italic m plus bold italic n bold i
    • We know that the complex conjugate bold italic m bold minus bold italic n bold i is another root
    • This means that z minus open parentheses m plus n straight i close parentheses and z minus open parentheses m minus n straight i close parenthesesare factors of the quadratic equation
    • Therefore z squared plus b z plus c equals left square bracket z minus open parentheses m plus n straight i close parentheses right square bracket left square bracket z minus open parentheses m minus n straight i close parentheses right square bracket
      • Writing this as the difference of two squares - left parenthesis open parentheses z minus m close parentheses minus n straight i right parenthesis left parenthesis open parentheses z minus m close parentheses plus n straight i right parenthesis - will speed up expanding
    • Expanding and simplifying gives us a quadratic equation where b and c are real numbers

Worked example

8-1-4-complex-roots-of-polynomials-example-solution-1-part-1

8-1-4-complex-roots-of-polynomials-example-solution-1-part-2

Examiner Tip

  • Once you have your final answers you can check your roots are correct by substituting your solutions back into the original equation.
  • You should get 0 if correct! [Note: 0 is equivalent to 0 plus 0 bold i]

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Complex Roots of Cubics & Quartics

How many roots should a polynomial have?

  • We know from previous sections that every quadratic equation has two roots (not necessarily distinct)
  • 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 a x cubed plus b x squared plus c x plus d equals 0 can have either:
      • 3 real roots
      • Or 1 real root and a complex conjugate pair
    • A quartic equation of the form a x to the power of 4 plus b x cubed plus c x squared plus d x plus e equals 0 will have the following cases for roots:
      • 4 real roots
      • 2 real and 2 non-real roots(a complex conjugate pair)
      • 4 non-real roots (two complex conjugate pairs)

Number of roots of a cubic function

Number of roots of a cubic function

Number of roots of a quartic 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 m plus n straight i is a root, then we know m minus n straight i is also a root
    • This means that left parenthesis z minus open parentheses m plus n straight i close parentheses right parenthesis and left parenthesis z minus left parenthesis m minus n i right parenthesis right parenthesis are factors of the cubic equation
    • Multiply the above factors together gives us a quadratic factor of the form left parenthesis A z squared plus B z plus C right parenthesis
    • We need to find the third factor left parenthesis z minus alpha right parenthesis
    • Multiply the factors and equate to our original equation to get

open parentheses A z squared plus B z plus C close parentheses open parentheses z minus alpha close parentheses equals a x cubed plus b x squared plus c x plus d

  • From there either
    • Expand and compare coefficients to find alpha
    • Or use polynomial division to find the factor open parentheses z minus alpha close parentheses
  • Finally, write your three roots clearly

How do we solve a quartic equation with complex roots?

  • When asked to find the roots of a quartic equation when we are given one, we use almost the same method as we did 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 left parenthesis z space minus space alpha right parenthesis and left parenthesis z space minus space beta right parenthesis
    • The unknown factors also form a quadratic factor left parenthesis z space minus space alpha right parenthesis left parenthesis z space minus space beta right parenthesis
    • 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 alpha and beta

How do we solve cubic/quartic equations with unknown coefficients?

  • Steps to find unknown variables in a given equation when given a root:
    • Substitute the given root into the equation space straight f left parenthesis z right parenthesis space equals space 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 either cubic/quartic equations

Worked example

8-1-4-complex-roots-of-polynomials-example-solution-2-part-a-part-1

8-1-4-complex-roots-of-polynomials-example-solution-2-part-a-part-2

Examiner Tip

  • As with solving quadratic equations, we can substitute our solutions back into the original equation to check we get 0.

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.