CIE A Level Maths: Pure 3复习笔记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

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  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 andare factors of the quadratic equation
    • Therefore
      • Writing this as 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

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

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

 

 

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