Learning Resources

Focus D: Roots of Polynomial Equations

Step B

• In Mathematical Modeling, Book 3, you discovered that a quadratic equation can be written in the following form:

x² - (sum of the 2 roots)x + (product of the 2 roots) = 0

• In this particular case, you are given that 4 - 5i is a root of a quadratic equation with integral coefficients.
• Any root of a quadratic equation with real coefficients that is a complex number has another root that is its conjugate.
• Thus, if one root is 4 - 5i, then the other root is 4 + 5i.
• The sum of these 2 roots is (4 - 5i ) + (4 + 5i ) = 8.
• The product of the roots is (4 - 5i )(4 + 5i ) = 16 - 25i ² = 41.
• Thus the desired quadratic equation is x² - 8x + 41 = 0.