Objective:

Sketch a polynomial that has a given number of real and complex roots, or determine the minimum number of real and complex roots from the graph, or vice versa

Sketch a polynomial with 2 real and 4 complex roots.

*What are real and complex roots?

-real roots: can be substituted in the equation to prove it true

-complex roots: aren't real solutions, part of it may be imaginary

*How do they look different when graphed?

-real roots: cross or touch the x-axis

-complex roots: do not cross or touch the x-axis

Sketch a polynomial with 4 real and 2 complex roots.

The polynomial will cross the x-axis 4 times because of the 4 real roots.

The polynomial has 2 curves that do not touch the x-axis because they are complex roots.

Sketch a polynomial with 5 real and 2 complex roots.

The polynomial will cross the x-axis 5 times because there are 5 real roots.

The polynomial curves twice, but doesn't touch the x-axis in these spots because they're complex roots.

Determine the minimum number of roots of this graph.

What should you look for when determining the minimum number of roots from a graph?

-turning points

-end behavior

Determine the minimum number of roots of this graph.

Turning points: 2

End behavior: going from left to right, graph points downwards and ends pointing up

What does this mean?

Because there are 2 turning points, the graph must have at least 2 roots. We know that a polynomial with odd degrees have ends pointing in different directions. In that case, the minimum number of roots for this graph is 3 ROOTS.

By looking at the graph, we can see that all roots touch the x-axis, which means there are no complex roots.