Math 120 Unit 2 – Functional Toolkit Part I Part B – Polynomial Functions

Activity 3 – Complex Roots Complex Numbers: the symbol “i” denotes imaginary numbers. i = √-1 i2 = (√-1)2 = -1 Simplify: √-9 = √9√-1 = ± 3√-1 = ± 3i

Roots of Quadratics: Given a quadratic equation ax2 + bx + c the roots can be found using the quadratic formula. Three types of roots can be found: 2 x-intercepts 1 x-intercept (double root) No x-intercept (complex roots) Complex roots always occur in pairs. Remember: complex roots occur when b2 - 4ac < 0.

Roots of Cubic's: Cubic equations have three roots. If we do not see three x-intercepts in the graph, then there are complex roots. Complex roots must always occur in pairs. Any graph can occur in reverse if a < 0.

3 distinct real roots 2 equal, 1 distinct, real roots local max local max local min local min 3 equal real roots 2 complex, 1 real roots local max No local min/max local min

Roots of Quartic's: Quartic equations have four roots. Many possible combinations of real and complex roots. Complex roots must always occur in pairs. Any graph can occur in reverse if a < 0.

4 distinct real roots 3 equal, 1 distinct, real roots local min local min absolute min absolute min 2 equal, 2 distinct real roots 2 pairs of equal real roots local min both min absolute min

4 equal real roots 2 distinct real, 2 complex roots local min absolute min absolute min 2 complex, 2 distinct real roots 2 distinct real, 2 complex roots local min absolute min absolute min

2 equal real, 2 complex roots local min absolute min absolute min 4 complex roots 2 minima

Graphing using Roots Determined the roots and place the real roots on the graph Determine if the real root crosses or touches the x-axis: Any even number of equal roots touch (not cross) the x-axis. Odd numbers of equal roots will pass through the x-axis. Place the complex roots in the graph. Complex roots may cause the graph to contain local maxima and minima. We can not yet determine their exact locations.

Example: Find the roots of the equation 0 = –(x – 2)(x2 – 5x + 10) then graph this equation. 1 real x-intercept: x = 2 Use the quadratic formula to find other roots:

Since a<0 then graph starts in 2nd quadrant and ends in 4th. y-intercept: y = –(0 – 2)(02 – 5(0) + 10) y = +20 (0, 20) (2, 0)