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Complex Roots of Polynomials (CIE A Level Maths: Pure 3)
Revision Note
Complex Roots of Quadratics
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 the difference of two squares - - will speed up expanding
- Expanding and simplifying gives us a quadratic equation where and are real numbers
Worked example
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 ]
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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 can have either:
- 3 real roots
- Or 1 real root and a complex conjugate pair
- A cubic equation of the form can have either:
-
- A quartic equation of the form 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)
- A quartic equation of the form will have the following cases for roots:
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 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 and
- The unknown factors also form a quadratic factor
- 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 and
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
- 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
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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