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

2. 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

3. 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.

4. 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.

5. 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

6. 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.

7. 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

8. 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

9. 2 equal real, 2 complex roots
local min
absolute min
absolute min
4 complex roots
2 minima

10. 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.

11. 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:

12. 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)