CIE A Level Maths: Pure 3复习笔记8.1.4 Complex Roots of Polynomials

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 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