CIE A Level Maths: Pure 3復習筆記8.1.4 Complex Roots of Polynomials
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
- If in the form
- 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
- If
- 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
- Writing this as
- Expanding?and?simplifying?gives us a quadratic equation where b and c?are?real?numbers
- We know that the complex conjugate
Worked Example
Exam 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 + 0i]
Complex Roots of Cubics & Quartics
How many roots should a polynomial have?
How do we solve a cubic equation with complex roots?
How do we solve a quartic equation with complex roots?
How do we solve cubic/quartic equations with unknown coefficients?
Worked Example
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